Theorem inn0f 4323

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 3920	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	exbii 1867	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3		inn0f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		inn0f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfin 4176	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
6	5	n0f 4301	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
7		df-rex 3086	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
8	2, 6, 7	3bitr4i 305	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1798   ∈ wcel 2141  Ⅎwnfc 2908   ≠ wne 2956  ∃wrex 3085   ∩ cin 3903  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-rex 3086  df-v 3455  df-dif 3907  df-in 3911  df-nul 4286

This theorem is referenced by:  inn0  4324