Theorem inn0f 4314

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 3911	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	exbii 1858	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3		inn0f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		inn0f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfin 4167	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
6	5	n0f 4292	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
7		df-rex 3077	. 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 398  ∃wex 1789   ∈ wcel 2132  Ⅎwnfc 2899   ≠ wne 2947  ∃wrex 3076   ∩ cin 3894  ∅c0 4276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-12 2202  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-rex 3077  df-v 3446  df-dif 3898  df-in 3902  df-nul 4277

This theorem is referenced by:  inn0  4315