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Mirrors > Home > MPE Home > Th. List > opabid | Structured version Visualization version GIF version |
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
opabid | ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5153 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
2 | copsexg 5176 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
3 | 2 | bicomd 215 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) |
4 | df-opab 4936 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
5 | 1, 3, 4 | elab2 3575 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1656 ∃wex 1878 ∈ wcel 2164 〈cop 4403 {copab 4935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4936 |
This theorem is referenced by: opelopabsb 5211 ssopab2b 5228 dmopab 5567 rnopab 5603 funopab 6158 opabiota 6508 fvopab5 6558 f1ompt 6630 ovid 7037 zfrep6 7396 enssdom 8247 omxpenlem 8330 infxpenlem 9149 canthwelem 9787 pospo 17326 2ndcdisj 21630 lgsquadlem1 25518 lgsquadlem2 25519 h2hlm 28381 opabdm 29959 opabrn 29960 fpwrelmap 30045 eulerpartlemgvv 30972 phpreu 33929 poimirlem26 33972 vvdifopab 34571 brabidga 34669 diclspsn 37262 areaquad 38637 sprsymrelf 42585 |
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