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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 5834 (this is a very limited reordering). | 
| Ref | Expression | 
|---|---|
| bj-0nelopab | ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relopab 5834 | . 2 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | 0nelrel0 5745 | . 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 2108 ∅c0 4333 {copab 5205 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: (None) | 
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