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Mirrors > Home > ILE Home > Th. List > opabid | 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 | vex 2604 | . . 3 ⊢ 𝑥 ∈ V | |
2 | vex 2604 | . . 3 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opex 3984 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V |
4 | copsexg 3999 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
5 | 4 | bicomd 139 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) |
6 | df-opab 3840 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
7 | 3, 5, 6 | elab2 2741 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 〈cop 3401 {copab 3838 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 |
This theorem is referenced by: opelopabsb 4015 ssopab2b 4031 dmopab 4564 rnopab 4599 funopab 4955 funco 4960 fvmptss2 5268 f1ompt 5341 ovid 5637 enssdom 6265 |
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