Theorem 0el 3473

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 2525	. 2 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∅)
2		eq0 3469	. . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3	2	rexbii 2504	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	1, 3	bitri 184	1 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  ∅c0 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by: (None)