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Mathbox for Peter Mazsa |
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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 1791 | . 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑 |
3 | opab0 5556 | . 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | |
4 | 2, 3 | mpbir 230 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1532 = wceq 1534 ∅c0 4323 {copab 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5211 |
This theorem is referenced by: coss0 37951 |
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