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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 37706 | . . . 4 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (¬ ∅ ∈ 𝐴 → dom (◡ E ↾ 𝐴) = 𝐴) |
3 | 2 | qseq1d 37663 | . 2 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = (𝐴 / (◡ E ↾ 𝐴))) |
4 | qsresid 37698 | . . 3 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = (𝐴 / ◡ E ) | |
5 | qsid 8774 | . . 3 ⊢ (𝐴 / ◡ E ) = 𝐴 | |
6 | 4, 5 | eqtri 2752 | . 2 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = 𝐴 |
7 | 3, 6 | eqtrdi 2780 | 1 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4315 E cep 5570 ◡ccnv 5666 dom cdm 5667 ↾ cres 5669 / cqs 8699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-eprel 5571 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 df-qs 8706 |
This theorem is referenced by: n0el3 38025 |
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