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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. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker opabidw 5529 when possible. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| opabid | ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opex 5469 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 2 | copsexg 5496 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
| 3 | 2 | bicomd 223 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) |
| 4 | df-opab 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 5 | 1, 3, 4 | elab2 3682 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-13 2377 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 |
| This theorem is referenced by: ssopab2b 5554 brabidga 38367 |
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