Theorem 0el 3431

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 2494	. 2 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = ∅)
2		eq0 3427	. . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3	2	rexbii 2473	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	1, 3	bitri 183	1 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  ∅c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by: (None)