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Mirrors > Home > MPE Home > Th. List > opab0 | Structured version Visualization version GIF version |
Description: Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
opab0 | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabn0 5405 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | |
2 | df-ne 2988 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅) | |
3 | 2exnaln 1830 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | |
4 | 1, 2, 3 | 3bitr3i 304 | . 2 ⊢ (¬ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) |
5 | 4 | con4bii 324 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 ≠ wne 2987 ∅c0 4243 {copab 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 |
This theorem is referenced by: iresn0n0 5890 fvmptopab 7188 epinid0 9048 cnvepnep 9055 opabf 35780 sprsymrelfvlem 44007 |
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