Theorem n0elim 39198

Description: Implication of that the empty set is not an element of a class. (Contributed by Peter Mazsa, 30-Dec-2024.)

Assertion

Ref	Expression
n0elim	⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)

Proof of Theorem n0elim

Step	Hyp	Ref	Expression
1		n0el2 38798	. . . 4 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)
2	1	biimpi 218	. . 3 ⊢ (¬ ∅ ∈ 𝐴 → dom (◡ E ↾ 𝐴) = 𝐴)
3	2	qseq1d 8736	. 2 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = (𝐴 / (◡ E ↾ 𝐴)))
4		qsresid 38794	. . 3 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = (𝐴 / ◡ E )
5		qsid 8758	. . 3 ⊢ (𝐴 / ◡ E ) = 𝐴
6	4, 5	eqtri 2784	. 2 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = 𝐴
7	3, 6	eqtrdi 2812	1 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1559   ∈ wcel 2141  ∅c0 4285   E cep 5544  ^◡ccnv 5644  dom cdm 5645   ↾ cres 5647   / cqs 8672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-eprel 5545  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ec 8675  df-qs 8679

This theorem is referenced by:  n0el3  39199