Theorem 0el 1720

Description: Membership of the empty set in another class.

Assertion

Ref	Expression
0el	⊢ (∅ ∈ A ↔ ∃x ∈ A ∀y ¬ y ∈ x)

Distinct variable group(s):   x,y,A

Proof of Theorem 0el

Step	Hyp	Ref	Expression
1		risset 1235	. 2 ⊢ (∅ ∈ A ↔ ∃x ∈ A x = ∅)
2		eq0 1719	. . 3 ⊢ (x = ∅ ↔ ∀y ¬ y ∈ x)
3	2	birex 1224	. 2 ⊢ (∃x ∈ A x = ∅ ↔ ∃x ∈ A ∀y ¬ y ∈ x)
4	1, 3	bitr 151	1 ⊢ (∅ ∈ A ↔ ∃x ∈ A ∀y ¬ y ∈ x)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∅c0 1707

This theorem is referenced by:  inf4 3473  zfinf 3474

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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