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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 5823 (this is a very limited reordering). |
Ref | Expression |
---|---|
bj-0nelopab | ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 5823 | . 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | 0nelrel0 5735 | . 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 2104 ∅c0 4321 {copab 5209 Rel wrel 5680 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-opab 5210 df-xp 5681 df-rel 5682 |
This theorem is referenced by: (None) |
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