Theorem inn0f 4346

Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
inn0f.1	⊢ Ⅎ𝑥𝐴
inn0f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
inn0f	⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Proof of Theorem inn0f

Step	Hyp	Ref	Expression
1		elin 3942	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	exbii 1848	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3		inn0f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		inn0f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfin 4199	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
6	5	n0f 4324	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
7		df-rex 3061	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
8	2, 6, 7	3bitr4i 303	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2108  Ⅎwnfc 2883   ≠ wne 2932  ∃wrex 3060   ∩ cin 3925  ∅c0 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-rex 3061  df-v 3461  df-dif 3929  df-in 3933  df-nul 4309

This theorem is referenced by:  inn0  4347