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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 5442 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | |
2 | df-ne 3019 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅) | |
3 | 2exnaln 1829 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | |
4 | 1, 2, 3 | 3bitr3i 303 | . 2 ⊢ (¬ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) |
5 | 4 | con4bii 323 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∀wal 1535 = wceq 1537 ∃wex 1780 ≠ wne 3018 ∅c0 4293 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 |
This theorem is referenced by: iresn0n0 5925 fvmptopab 7211 epinid0 9066 cnvepnep 9073 opabf 35622 sprsymrelfvlem 43659 |
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