Theorem n0el3 36763

Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.)

Assertion

Ref	Expression
n0el3	⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)

Proof of Theorem n0el3

Step	Hyp	Ref	Expression
1		n0el2 36468	. . . . 5 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)
2	1	biimpi 215	. . . 4 ⊢ (¬ ∅ ∈ 𝐴 → dom (◡ E ↾ 𝐴) = 𝐴)
3	2	qseq1d 36425	. . 3 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = (𝐴 / (◡ E ↾ 𝐴)))
4		qsresid 36460	. . . 4 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = (𝐴 / ◡ E )
5		qsid 8572	. . . 4 ⊢ (𝐴 / ◡ E ) = 𝐴
6	4, 5	eqtri 2766	. . 3 ⊢ (𝐴 / (◡ E ↾ 𝐴)) = 𝐴
7	3, 6	eqtrdi 2794	. 2 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
8		n0eldmqseq 36762	. 2 ⊢ ((dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴 → ¬ ∅ ∈ 𝐴)
9	7, 8	impbii 208	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∅c0 4256   E cep 5494  ^◡ccnv 5588  dom cdm 5589   ↾ cres 5591   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by:  cnvepresdmqss  36764  cnvepresdmqs  36765