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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-0nelopab | Structured version Visualization version GIF version |
Description: The empty set is never an
element in an ordered-pair class abstraction.
(Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened
by BJ, 22-Jul-2023.)
TODO: move to the main section when one can reorder sections so that we can use relopab 5837 (this is a very limited reordering). |
Ref | Expression |
---|---|
bj-0nelopab | ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 5837 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 0nelrel0 5749 | . 2 ⊢ (Rel {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 ∅c0 4339 {copab 5210 Rel wrel 5694 |
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-12 2175 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-rab 3434 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 df-xp 5695 df-rel 5696 |
This theorem is referenced by: (None) |
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