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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2finon 42201 | 2 is a finite ordinal. See 1onn 8639. (Contributed by RP, 27-Sep-2023.) |
⊢ 2o ∈ (On ∩ Fin) | ||
Theorem | 3finon 42202 | 3 is a finite ordinal. See 1onn 8639. (Contributed by RP, 27-Sep-2023.) |
⊢ 3o ∈ (On ∩ Fin) | ||
Theorem | 4finon 42203 | 4 is a finite ordinal. See 1onn 8639. (Contributed by RP, 27-Sep-2023.) |
⊢ 4o ∈ (On ∩ Fin) | ||
Theorem | finona1cl 42204 | The finite ordinals are closed under the add one operation. (Contributed by RP, 27-Sep-2023.) |
⊢ (𝑁 ∈ (On ∩ Fin) → (𝑁 +o 1o) ∈ (On ∩ Fin)) | ||
Theorem | finonex 42205 | The finite ordinals are a set. See also onprc 7765 and fiprc 9045 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.) |
⊢ (On ∩ Fin) ∈ V | ||
Theorem | fzunt 42206 | Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.) |
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | fzuntd 42207 | Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | fzunt1d 42208 | Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ 𝐿) & ⊢ (𝜑 → 𝐿 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | fzuntgd 42209 | Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.) |
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ (𝐿 + 1)) & ⊢ (𝜑 → 𝐿 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) | ||
Theorem | ifpan123g 42210 | Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂)))) | ||
Theorem | ifpan23 42211 | Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.) |
⊢ ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓 ∧ 𝜃), (𝜒 ∧ 𝜏))) | ||
Theorem | ifpdfor2 42212 | Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓)) | ||
Theorem | ifporcor 42213 | Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.) |
⊢ (if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑)) | ||
Theorem | ifpdfan2 42214 | Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, 𝜑)) | ||
Theorem | ifpancor 42215 | Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓)) | ||
Theorem | ifpdfor 42216 | Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓)) | ||
Theorem | ifpdfan 42217 | Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥)) | ||
Theorem | ifpbi2 42218 | Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃))) | ||
Theorem | ifpbi3 42219 | Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓))) | ||
Theorem | ifpim1 42220 | Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓)) | ||
Theorem | ifpnot 42221 | Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ (¬ 𝜑 ↔ if-(𝜑, ⊥, ⊤)) | ||
Theorem | ifpid2 42222 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥)) | ||
Theorem | ifpim2 42223 | Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑)) | ||
Theorem | ifpbi23 42224 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃))) | ||
Theorem | ifpbiidcor 42225 | Restatement of biid 261. (Contributed by RP, 25-Apr-2020.) |
⊢ if-(𝜑, 𝜑, ¬ 𝜑) | ||
Theorem | ifpbicor 42226 | Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | ifpxorcor 42227 | Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑)) | ||
Theorem | ifpbi1 42228 | Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃))) | ||
Theorem | ifpnot23 42229 | Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.) |
⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒)) | ||
Theorem | ifpnotnotb 42230 | Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | ifpnorcor 42231 | Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑)) | ||
Theorem | ifpnancor 42232 | Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓)) | ||
Theorem | ifpnot23b 42233 | Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒)) | ||
Theorem | ifpbiidcor2 42234 | Restatement of biid 261. (Contributed by RP, 25-Apr-2020.) |
⊢ ¬ if-(𝜑, ¬ 𝜑, 𝜑) | ||
Theorem | ifpnot23c 42235 | Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒)) | ||
Theorem | ifpnot23d 42236 | Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | ifpdfnan 42237 | Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ if-(𝜑, ¬ 𝜓, ⊤)) | ||
Theorem | ifpdfxor 42238 | Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜓)) | ||
Theorem | ifpbi12 42239 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏))) | ||
Theorem | ifpbi13 42240 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃))) | ||
Theorem | ifpbi123 42241 | Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂))) | ||
Theorem | ifpidg 42242 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜃 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑 ∧ 𝜓) → 𝜃) ∧ ((𝜑 ∧ 𝜃) → 𝜓)) ∧ ((𝜒 → (𝜑 ∨ 𝜃)) ∧ (𝜃 → (𝜑 ∨ 𝜒))))) | ||
Theorem | ifpid3g 42243 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓))) | ||
Theorem | ifpid2g 42244 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑 ∨ 𝜒)) ∧ (𝜒 → (𝜑 ∨ 𝜓)))) | ||
Theorem | ifpid1g 42245 | Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒 → 𝜑) ∧ (𝜑 → 𝜓))) | ||
Theorem | ifpim23g 42246 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (((𝜑 → 𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ (𝜒 → (𝜑 ∨ 𝜓)))) | ||
Theorem | ifpim3 42247 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑)) | ||
Theorem | ifpnim1 42248 | Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑)) | ||
Theorem | ifpim4 42249 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑)) | ||
Theorem | ifpnim2 42250 | Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑)) | ||
Theorem | ifpim123g 42251 | Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 → 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 → 𝜂))))) | ||
Theorem | ifpim1g 42252 | Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)))) | ||
Theorem | ifp1bi 42253 | Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((𝜑 → 𝜓) ∨ (𝜃 → 𝜒))) ∧ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒))))) | ||
Theorem | ifpbi1b 42254 | When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.) |
⊢ (if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒)) | ||
Theorem | ifpimimb 42255 | Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpororb 42256 | Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ∨ 𝜒), (𝜃 ∨ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∨ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpananb 42257 | Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ∧ 𝜒), (𝜃 ∧ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpnannanb 42258 | Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ⊼ 𝜒), (𝜃 ⊼ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊼ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpor123g 42259 | Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂))))) | ||
Theorem | ifpimim 42260 | Consequnce of implication. (Contributed by RP, 17-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpbibib 42261 | Factor conditional logic operator over biconditional in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ↔ 𝜒), (𝜃 ↔ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | ifpxorxorb 42262 | Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
⊢ (if-(𝜑, (𝜓 ⊻ 𝜒), (𝜃 ⊻ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊻ if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | rp-fakeimass 42263 | A special case where implication appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
⊢ ((𝜑 ∨ 𝜒) ↔ (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))) | ||
Theorem | rp-fakeanorass 42264 | A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.) |
⊢ ((𝜒 → 𝜑) ↔ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒)))) | ||
Theorem | rp-fakeoranass 42265 | A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
⊢ ((𝜑 → 𝜒) ↔ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒)))) | ||
Theorem | rp-fakeinunass 42266 | A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.) |
⊢ (𝐶 ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶))) | ||
Theorem | rp-fakeuninass 42267 | A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶))) | ||
Membership in the class of finite sets can be expressed in many ways. | ||
Theorem | rp-isfinite5 42268* | A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ0. (Contributed by RP, 3-Mar-2020.) |
⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴) | ||
Theorem | rp-isfinite6 42269* | A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ. (Contributed by RP, 10-Mar-2020.) |
⊢ (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)) | ||
Theorem | intabssd 42270* | When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑦) ⇒ ⊢ (𝜑 → ∩ {𝑥 ∣ 𝜓} ⊆ ∩ {𝑦 ∣ 𝜒}) | ||
Theorem | eu0 42271* | There is only one empty set. (Contributed by RP, 1-Oct-2023.) |
⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | ||
Theorem | epelon2 42272 | Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴 ∈ 𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7762. This is a weak form of epelg 5582 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | ontric3g 42273* | For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥))) | ||
Theorem | dfsucon 42274* | 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.) |
⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥) | ||
Theorem | snen1g 42275 | A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.) |
⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) | ||
Theorem | snen1el 42276 | A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.) |
⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ {𝐴}) | ||
Theorem | sn1dom 42277 | A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.) |
⊢ {𝐴} ≼ 1o | ||
Theorem | pr2dom 42278 | An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.) |
⊢ {𝐴, 𝐵} ≼ 2o | ||
Theorem | tr3dom 42279 | An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.) |
⊢ {𝐴, 𝐵, 𝐶} ≼ 3o | ||
Theorem | ensucne0 42280 | A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.) |
⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅) | ||
Theorem | ensucne0OLD 42281 | A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅) | ||
Theorem | dfom6 42282 | Let ω be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023.) |
⊢ ω = ∪ (On ∩ Fin) | ||
Theorem | infordmin 42283 | ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.) |
⊢ ∀𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥 | ||
Theorem | iscard4 42284 | Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.) |
⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card) | ||
Theorem | minregex 42285* | Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of ℵ which is greater to or equal to 𝐴. This proof uses AC. (Contributed by RP, 23-Nov-2023.) |
⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))}) | ||
Theorem | minregex2 42286* | Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of ℵ which dominates 𝐴. This proof uses AC. (Contributed by RP, 24-Nov-2023.) |
⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))}) | ||
Theorem | iscard5 42287* | Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.) |
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) | ||
Theorem | elrncard 42288* | Let us define a cardinal number to be an element 𝐴 ∈ On such that 𝐴 is not equipotent with any 𝑥 ∈ 𝐴. (Contributed by RP, 1-Oct-2023.) |
⊢ (𝐴 ∈ ran card ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) | ||
Theorem | harval3 42289* | (har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.) |
⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥}) | ||
Theorem | harval3on 42290* | For any ordinal number 𝐴 let (har‘𝐴) denote the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.) |
⊢ (𝐴 ∈ On → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥}) | ||
Theorem | omssrncard 42291 | All natural numbers are cardinals. (Contributed by RP, 1-Oct-2023.) |
⊢ ω ⊆ ran card | ||
Theorem | 0iscard 42292 | 0 is a cardinal number. (Contributed by RP, 1-Oct-2023.) |
⊢ ∅ ∈ ran card | ||
Theorem | 1iscard 42293 | 1 is a cardinal number. (Contributed by RP, 1-Oct-2023.) |
⊢ 1o ∈ ran card | ||
Theorem | omiscard 42294 | ω is a cardinal number. (Contributed by RP, 1-Oct-2023.) |
⊢ ω ∈ ran card | ||
Theorem | sucomisnotcard 42295 | ω +o 1o is not a cardinal number. (Contributed by RP, 1-Oct-2023.) |
⊢ ¬ (ω +o 1o) ∈ ran card | ||
Theorem | nna1iscard 42296 | For any natural number, the add one operation is results in a cardinal number. (Contributed by RP, 1-Oct-2023.) |
⊢ (𝑁 ∈ ω → (𝑁 +o 1o) ∈ ran card) | ||
Theorem | har2o 42297 | The least cardinal greater than 2 is 3. (Contributed by RP, 5-Nov-2023.) |
⊢ (har‘2o) = 3o | ||
Theorem | en2pr 42298* | A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.) |
⊢ (𝐴 ≈ 2o ↔ ∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦)) | ||
Theorem | pr2cv 42299 | If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.) |
⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | pr2el1 42300 | If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.) |
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ {𝐴, 𝐵}) |
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