Theorem 0el 4308

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 3207	. 2 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∅)
2		eq0 4295	. . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3	2	rexbii 3079	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	1, 3	bitri 275	1 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ∅c0 4278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-dif 3900  df-nul 4279

This theorem is referenced by:  n0el  4309  axinf2  9525  zfinf2  9527  gneispace  44167