Theorem 0el 3566

Description: Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Assertion

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

Distinct variable groups:   x,A   x,y

Allowed substitution hint:   A(y)

Proof of Theorem 0el

Step	Hyp	Ref	Expression
1		risset 2661	. 2 ⊢ (∅ ∈ A ↔ ∃x ∈ A x = ∅)
2		eq0 3564	. . 3 ⊢ (x = ∅ ↔ ∀y ¬ y ∈ x)
3	2	rexbii 2639	. 2 ⊢ (∃x ∈ A x = ∅ ↔ ∃x ∈ A ∀y ¬ y ∈ x)
4	1, 3	bitri 240	1 ⊢ (∅ ∈ A ↔ ∃x ∈ A ∀y ¬ y ∈ x)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∅c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by: (None)