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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > n0elim | Structured version Visualization version GIF version |
Description: Implication of that the empty set is not an element of a class. (Contributed by Peter Mazsa, 30-Dec-2024.) |
Ref | Expression |
---|---|
n0elim | ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0el2 37805 | . . . 4 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (¬ ∅ ∈ 𝐴 → dom (◡ E ↾ 𝐴) = 𝐴) |
3 | 2 | qseq1d 8783 | . 2 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = (𝐴 / (◡ E ↾ 𝐴))) |
4 | qsresid 37797 | . . 3 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = (𝐴 / ◡ E ) | |
5 | qsid 8802 | . . 3 ⊢ (𝐴 / ◡ E ) = 𝐴 | |
6 | 4, 5 | eqtri 2756 | . 2 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = 𝐴 |
7 | 3, 6 | eqtrdi 2784 | 1 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∅c0 4323 E cep 5581 ◡ccnv 5677 dom cdm 5678 ↾ cres 5680 / cqs 8724 |
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-10 2130 ax-11 2147 ax-12 2167 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-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 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-br 5149 df-opab 5211 df-eprel 5582 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 df-qs 8731 |
This theorem is referenced by: n0el3 38123 |
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