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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ensymd 9001 | Symmetry of equinumerosity. Deduction form of ensym 8999. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≈ 𝐴) | ||
Theorem | entr 9002 | Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | domtr 9003 | Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | entri 9004 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | entr2i 9005 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐶 ≈ 𝐴 | ||
Theorem | entr3i 9006 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐴 ≈ 𝐶 ⇒ ⊢ 𝐵 ≈ 𝐶 | ||
Theorem | entr4i 9007 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐶 ≈ 𝐵 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | endomtr 9008 | Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | domentr 9009 | Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | f1imaeng 9010 | If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | f1imaen2g 9011 | If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (This version of f1imaeng 9010 does not need ax-rep 5286.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | f1imaen 9012 | If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.) |
⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | en0 9013 | The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5364, ax-un 7725. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | ||
Theorem | en0OLD 9014 | Obsolete version of en0 9013 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5364. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | ||
Theorem | en0ALT 9015 | Shorter proof of en0 9013, depending on ax-pow 5364 and ax-un 7725. (Contributed by NM, 27-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | ||
Theorem | en0r 9016 | The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.) |
⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) | ||
Theorem | ensn1 9017 | A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7725. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ≈ 1o | ||
Theorem | ensn1OLD 9018 | Obsolete version of ensn1 9017 as of 23-Sep-2024. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ≈ 1o | ||
Theorem | ensn1g 9019 | A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | ||
Theorem | enpr1g 9020 | {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) | ||
Theorem | en1 9021* | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7725. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | en1OLD 9022* | Obsolete version of en1 9021 as of 23-Sep-2024. (Contributed by NM, 25-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | en1b 9023 | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7725. (Revised by BTernaryTau, 24-Sep-2024.) |
⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) | ||
Theorem | en1bOLD 9024 | Obsolete version of en1b 9023 as of 24-Sep-2024. (Contributed by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) | ||
Theorem | reuen1 9025* | Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o) | ||
Theorem | euen1 9026 | Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) | ||
Theorem | euen1b 9027* | Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.) |
⊢ (𝐴 ≈ 1o ↔ ∃!𝑥 𝑥 ∈ 𝐴) | ||
Theorem | en1uniel 9028 | A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7725. (Revised by BTernaryTau, 24-Sep-2024.) |
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) | ||
Theorem | en1unielOLD 9029 | Obsolete version of en1uniel 9028 as of 24-Sep-2024. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) | ||
Theorem | 2dom 9030* | A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.) |
⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) | ||
Theorem | fundmen 9031 | A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹) | ||
Theorem | fundmeng 9032 | A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹) | ||
Theorem | cnven 9033 | A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴) | ||
Theorem | cnvct 9034 | If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) | ||
Theorem | fndmeng 9035 | A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹) | ||
Theorem | mapsnend 9036 | Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴) | ||
Theorem | mapsnen 9037 | Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑m {𝐵}) ≈ 𝐴 | ||
Theorem | snmapen 9038 | Set exponentiation: a singleton to any set is equinumerous to that singleton. (Contributed by NM, 17-Dec-2003.) (Revised by AV, 17-Jul-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴}) | ||
Theorem | snmapen1 9039 | Set exponentiation: a singleton to any set is equinumerous to ordinal 1. (Proposed by BJ, 17-Jul-2022.) (Contributed by AV, 17-Jul-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ 1o) | ||
Theorem | map1 9040 | Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) | ||
Theorem | en2sn 9041 | Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5364. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7725. (Revised by BTernaryTau, 25-Sep-2024.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) | ||
Theorem | en2snOLD 9042 | Obsolete version of en2sn 9041 as of 25-Sep-2024. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5364. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) | ||
Theorem | en2snOLDOLD 9043 | Obsolete version of en2sn 9041 as of 31-Jul-2024. (Contributed by NM, 9-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) | ||
Theorem | snfi 9044 | A singleton is finite. (Contributed by NM, 4-Nov-2002.) |
⊢ {𝐴} ∈ Fin | ||
Theorem | fiprc 9045 | The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.) |
⊢ Fin ∉ V | ||
Theorem | unen 9046 | Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)) | ||
Theorem | enrefnn 9047 | Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8980). (Contributed by BTernaryTau, 31-Jul-2024.) |
⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | ||
Theorem | en2prd 9048 | Two unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷}) | ||
Theorem | enpr2d 9049 | A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-un 7725. (Revised by BTernaryTau, 23-Dec-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | enpr2dOLD 9050 | Obsolete version of enpr2d 9049 as of 23-Dec-2024. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | ssct 9051 | Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) Avoid ax-pow 5364, ax-un 7725. (Revised by BTernaryTau, 7-Dec-2024.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | ||
Theorem | ssctOLD 9052 | Obsolete version of ssct 9051 as of 7-Dec-2024. (Contributed by Thierry Arnoux, 31-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | ||
Theorem | difsnen 9053 | All decrements of a set are equinumerous. (Contributed by Stefan O'Rear, 19-Feb-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵})) | ||
Theorem | domdifsn 9054 | Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶})) | ||
Theorem | xpsnen 9055 | A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 × {𝐵}) ≈ 𝐴 | ||
Theorem | xpsneng 9056 | A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴) | ||
Theorem | xp1en 9057 | One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1o) ≈ 𝐴) | ||
Theorem | endisj 9058* | Any two sets are equinumerous to two disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) | ||
Theorem | undom 9059 | Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5364. (Revised by BTernaryTau, 4-Dec-2024.) |
⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷)) | ||
Theorem | undomOLD 9060 | Obsolete version of undom 9059 as of 4-Dec-2024. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷)) | ||
Theorem | xpcomf1o 9061* | The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.) |
⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) ⇒ ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) | ||
Theorem | xpcomco 9062* | Composition with the bijection of xpcomf1o 9061 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥}) & ⊢ 𝐺 = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶) ⇒ ⊢ (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | ||
Theorem | xpcomen 9063 | Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴) | ||
Theorem | xpcomeng 9064 | Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)) | ||
Theorem | xpsnen2g 9065 | A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ 𝐵) | ||
Theorem | xpassen 9066 | Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶)) | ||
Theorem | xpdom2 9067 | Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)) | ||
Theorem | xpdom2g 9068 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)) | ||
Theorem | xpdom1g 9069 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | ||
Theorem | xpdom3 9070 | A set is dominated by its Cartesian product with a nonempty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵)) | ||
Theorem | xpdom1 9071 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | ||
Theorem | domunsncan 9072 | A singleton cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋 ≼ 𝑌)) | ||
Theorem | omxpenlem 9073* | Lemma for omxpen 9074. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵)) | ||
Theorem | omxpen 9074 | The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ≈ (𝐴 × 𝐵)) | ||
Theorem | omf1o 9075* | Construct an explicit bijection from 𝐴 ·o 𝐵 to 𝐵 ·o 𝐴. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) & ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴)) | ||
Theorem | pw2f1olem 9076* | Lemma for pw2f1o 9077. (Contributed by Mario Carneiro, 6-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶})))) | ||
Theorem | pw2f1o 9077* | The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ⇒ ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴)) | ||
Theorem | pw2eng 9078 | The power set of a set is equinumerous to set exponentiation with a base of ordinal 2o. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) | ||
Theorem | pw2en 9079 | The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ≈ (2o ↑m 𝐴) | ||
Theorem | fopwdom 9080 | Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴) | ||
Theorem | enfixsn 9081* | Given two equipollent sets, a bijection can always be chosen which fixes a single point. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵)) | ||
Theorem | sucdom2OLD 9082 | Obsolete version of sucdom2 9206 as of 4-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) | ||
Theorem | sbthlem1 9083* | Lemma for sbth 9093. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) | ||
Theorem | sbthlem2 9084* | Lemma for sbth 9093. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷) | ||
Theorem | sbthlem3 9085* | Lemma for sbth 9093. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷)) | ||
Theorem | sbthlem4 9086* | Lemma for sbth 9093. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) | ||
Theorem | sbthlem5 9087* | Lemma for sbth 9093. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = 𝐴) | ||
Theorem | sbthlem6 9088* | Lemma for sbth 9093. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵) | ||
Theorem | sbthlem7 9089* | Lemma for sbth 9093. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻) | ||
Theorem | sbthlem8 9090* | Lemma for sbth 9093. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻) | ||
Theorem | sbthlem9 9091* | Lemma for sbth 9093. (Contributed by NM, 28-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) | ||
Theorem | sbthlem10 9092* | Lemma for sbth 9093. (Contributed by NM, 28-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | ||
Theorem | sbth 9093 |
Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This
theorem states that if set 𝐴 is smaller (has lower cardinality)
than
𝐵 and vice-versa, then 𝐴 and
𝐵
are equinumerous (have the
same cardinality). The interesting thing is that this can be proved
without invoking the Axiom of Choice, as we do here. The theorem can
also be proved from the axiom of choice and the linear order of the
cardinal numbers, but our development does not provide the linear order
of cardinal numbers until much later and in ways that depend on
Schroeder-Bernstein.
The main proof consists of lemmas sbthlem1 9083 through sbthlem10 9092; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 9092. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. In the Intuitionistic Logic Explorer (ILE) the Schroeder-Bernstein Theorem has been proven equivalent to the law of the excluded middle (LEM), and in ILE the LEM is not accepted as necessarily true; see https://us.metamath.org/ileuni/exmidsbth.html 9092. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | ||
Theorem | sbthb 9094 | Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵) | ||
Theorem | sbthcl 9095 | Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.) |
⊢ ≈ = ( ≼ ∩ ◡ ≼ ) | ||
Theorem | dfsdom2 9096 | Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | ||
Theorem | brsdom2 9097 | Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴)) | ||
Theorem | sdomnsym 9098 | Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | ||
Theorem | domnsym 9099 | Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | ||
Theorem | 0domg 9100 | Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5364, ax-un 7725. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) |
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