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Mirrors > Home > MPE Home > Th. List > opabidw | Structured version Visualization version GIF version |
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5413 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 14-Apr-1995.) (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
opabidw | ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5356 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
2 | copsexgw 5381 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
3 | 2 | bicomd 225 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) |
4 | df-opab 5129 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
5 | 1, 3, 4 | elab2 3670 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4573 {copab 5128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 |
This theorem is referenced by: rexopabb 5415 opelopabsb 5417 ssopab2bw 5434 dmopab 5784 rnopab 5826 funopab 6390 opabiota 6746 fvopab5 6800 f1ompt 6875 ovid 7291 zfrep6 7656 enssdom 8534 omxpenlem 8618 infxpenlem 9439 canthwelem 10072 pospo 17583 2ndcdisj 22064 lgsquadlem1 25956 lgsquadlem2 25957 h2hlm 28757 opabdm 30362 opabrn 30363 fpwrelmap 30469 eulerpartlemgvv 31634 satfvsucsuc 32612 phpreu 34891 poimirlem26 34933 vvdifopab 35536 brabidgaw 35632 diclspsn 38345 areaquad 39872 sprsymrelf 43706 |
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