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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 5511 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | |
2 | df-ne 2945 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅) | |
3 | 2exnaln 1832 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | |
4 | 1, 2, 3 | 3bitr3i 301 | . 2 ⊢ (¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) |
5 | 4 | con4bii 321 | 1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ≠ wne 2944 ∅c0 4283 {copab 5168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 |
This theorem is referenced by: iresn0n0 6008 fvmptopabOLD 7413 epinid0 9537 cnvepnep 9545 opabf 36832 sprsymrelfvlem 45689 |
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