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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 2684 | . . 3 ⊢ 𝑥 ∈ V | |
2 | vex 2684 | . . 3 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opex 4146 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V |
4 | copsexg 4161 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
5 | 4 | bicomd 140 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) |
6 | df-opab 3985 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
7 | 3, 5, 6 | elab2 2827 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 〈cop 3525 {copab 3983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 |
This theorem is referenced by: opelopabsb 4177 ssopab2b 4193 dmopab 4745 rnopab 4781 funopab 5153 funco 5158 fvmptss2 5489 f1ompt 5564 ovid 5880 enssdom 6649 |
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