Theorem n0el3 39240

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		n0elim 39239	. 2 ⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
2		n0eldmqseq 39238	. 2 ⊢ ((dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴 → ¬ ∅ ∈ 𝐴)
3	1, 2	impbii 211	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1562   ∈ wcel 2144  ∅c0 4287   E cep 5548  ^◡ccnv 5648  dom cdm 5649   ↾ cres 5651   / cqs 8679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-eprel 5549  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-qs 8686

This theorem is referenced by:  cnvepresdmqss  39241  cnvepresdmqs  39242  eldisjn0elb  39349  eldisjn0el  39413