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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opabf | Structured version Visualization version GIF version | ||
| Description: A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.) (Proof shortened by Thierry Arnoux, 18-Feb-2022.) | 
| Ref | Expression | 
|---|---|
| opabf.1 | ⊢ ¬ 𝜑 | 
| Ref | Expression | 
|---|---|
| opabf | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opabf.1 | . . 3 ⊢ ¬ 𝜑 | |
| 2 | 1 | gen2 1795 | . 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑 | 
| 3 | opab0 5558 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | |
| 4 | 2, 3 | mpbir 231 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∀wal 1537 = wceq 1539 ∅c0 4332 {copab 5204 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 | 
| This theorem is referenced by: coss0 38481 | 
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