![]() |
Metamath
Proof Explorer Theorem List (p. 41 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ssrabdv 4001* | Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓) ⇒ ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) | ||
Theorem | rabssdv 4002* | Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵) | ||
Theorem | ss2rabdv 4003* | Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | ss2rabi 4004 | Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabss2 4005* | Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑}) | ||
Theorem | ssab2 4006* | Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.) |
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | ||
Theorem | ssrab2 4007* | Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | ||
Theorem | ssrab3 4008* | Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} ⇒ ⊢ 𝐵 ⊆ 𝐴 | ||
Theorem | rabssrabd 4009* | Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | ssrabeq 4010* | If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) | ||
Theorem | rabssab 4011 | A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | ||
Theorem | uniiunlem 4012* | A subset relationship useful for converting union to indexed union using dfiun2 4920 or dfiun2g 4917 and intersection to indexed intersection using dfiin2 4921. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)) | ||
Theorem | dfpss2 4013 | Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) |
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | ||
Theorem | dfpss3 4014 | Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | ||
Theorem | psseq1 4015 | Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) |
⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | ||
Theorem | psseq2 4016 | Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) |
⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) | ||
Theorem | psseq1i 4017 | An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶) | ||
Theorem | psseq2i 4018 | An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵) | ||
Theorem | psseq12i 4019 | An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) | ||
Theorem | psseq1d 4020 | An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | ||
Theorem | psseq2d 4021 | An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) | ||
Theorem | psseq12d 4022 | An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) | ||
Theorem | pssss 4023 | A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | pssne 4024 | Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) | ||
Theorem | pssssd 4025 | Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | pssned 4026 | Proper subclasses are unequal. Deduction form of pssne 4024. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | sspss 4027 | Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | pssirr 4028 | Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
⊢ ¬ 𝐴 ⊊ 𝐴 | ||
Theorem | pssn2lp 4029 | Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴) | ||
Theorem | sspsstri 4030 | Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)) | ||
Theorem | ssnpss 4031 | Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴) | ||
Theorem | psstr 4032 | Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | sspsstr 4033 | Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | psssstr 4034 | Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | psstrd 4035 | Proper subclass inclusion is transitive. Deduction form of psstr 4032. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) & ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | sspsstrd 4036 | Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4033. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | psssstrd 4037 | Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4034. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | npss 4038 | A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3930. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) | ||
Theorem | ssnelpss 4039 | A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) | ||
Theorem | ssnelpssd 4040 | Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4039. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | ||
Theorem | ssexnelpss 4041* | If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) | ||
Theorem | dfdif3 4042* | Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) |
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦} | ||
Theorem | difeq1 4043 | Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | ||
Theorem | difeq2 4044 | Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | ||
Theorem | difeq12 4045 | Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | ||
Theorem | difeq1i 4046 | Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) | ||
Theorem | difeq2i 4047 | Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵) | ||
Theorem | difeq12i 4048 | Equality inference for class difference. (Contributed by NM, 29-Aug-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) | ||
Theorem | difeq1d 4049 | Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | ||
Theorem | difeq2d 4050 | Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | ||
Theorem | difeq12d 4051 | Equality deduction for class difference. (Contributed by FL, 29-May-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | ||
Theorem | difeqri 4052* | Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∖ 𝐵) = 𝐶 | ||
Theorem | nfdif 4053 | Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) | ||
Theorem | eldifi 4054 | Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | eldifn 4055 | Implication of membership in a class difference. (Contributed by NM, 3-May-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶) | ||
Theorem | elndif 4056 | A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) | ||
Theorem | neldif 4057 | Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶) | ||
Theorem | difdif 4058 | Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.) |
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | ||
Theorem | difss 4059 | Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | ||
Theorem | difssd 4060 | A difference of two classes is contained in the minuend. Deduction form of difss 4059. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | ||
Theorem | difss2 4061 | If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) | ||
Theorem | difss2d 4062 | If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4061. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | ssdifss 4063 | Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | ddif 4064 | Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | ||
Theorem | ssconb 4065 | Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴))) | ||
Theorem | sscon 4066 | Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | ||
Theorem | ssdif 4067 | Difference law for subsets. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | ssdifd 4068 | If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4067. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | sscond 4069 | If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4066. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | ||
Theorem | ssdifssd 4070 | If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 4063. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | ssdif2d 4071 | If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | raldifb 4072 | Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑) | ||
Theorem | rexdifi 4073 | Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.) |
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑) | ||
Theorem | complss 4074 | Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | ||
Theorem | compleq 4075 | Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.) |
⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵)) | ||
Theorem | elun 4076 | Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | ||
Theorem | elunnel1 4077 | A member of a union that is not member of the first class, is member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) | ||
Theorem | uneqri 4078* | Inference from membership to union. (Contributed by NM, 21-Jun-1993.) |
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∪ 𝐵) = 𝐶 | ||
Theorem | unidm 4079 | Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 ∪ 𝐴) = 𝐴 | ||
Theorem | uncom 4080 | Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | ||
Theorem | equncom 4081 | If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4081 was automatically derived from equncomVD 41574 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) | ||
Theorem | equncomi 4082 | Inference form of equncom 4081. equncomi 4082 was automatically derived from equncomiVD 41575 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) | ||
Theorem | uneq1 4083 | Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | ||
Theorem | uneq2 4084 | Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | ||
Theorem | uneq12 4085 | Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | ||
Theorem | uneq1i 4086 | Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) | ||
Theorem | uneq2i 4087 | Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) | ||
Theorem | uneq12i 4088 | Equality inference for the union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) | ||
Theorem | uneq1d 4089 | Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | ||
Theorem | uneq2d 4090 | Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | ||
Theorem | uneq12d 4091 | Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | ||
Theorem | nfun 4092 | Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) | ||
Theorem | unass 4093 | Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | ||
Theorem | un12 4094 | A rearrangement of union. (Contributed by NM, 12-Aug-2004.) |
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | ||
Theorem | un23 4095 | A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | ||
Theorem | un4 4096 | A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) | ||
Theorem | unundi 4097 | Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) | ||
Theorem | unundir 4098 | Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) | ||
Theorem | ssun1 4099 | Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | ||
Theorem | ssun2 4100 | Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |