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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 1793 | . 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑 |
3 | opab0 5564 | . 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 1535 = wceq 1537 ∅c0 4339 {copab 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 |
This theorem is referenced by: coss0 38461 |
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