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Mirrors > Home > MPE Home > Th. List > opelopab | Structured version Visualization version GIF version |
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
Ref | Expression |
---|---|
opelopab.1 | ⊢ 𝐴 ∈ V |
opelopab.2 | ⊢ 𝐵 ∈ V |
opelopab.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopab.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opelopab | ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopab.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelopab.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opelopab.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | opelopab.4 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | opelopabg 5220 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
6 | 1, 2, 5 | mp2an 685 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 Vcvv 3415 〈cop 4404 {copab 4936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4937 |
This theorem is referenced by: opabid2 5485 dfres2 5691 f1oiso 6857 elopabi 7495 xporderlem 7553 cnlnssadj 29495 areacirclem5 34048 dicopelval 37253 dih1dimatlem 37405 pellexlem3 38240 fsovrfovd 39144 |
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