Theorem inn0f 39763

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 3947	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	exbii 1924	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3		inn0f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		inn0f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
5	3, 4	nfin 3969	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
6	5	n0f 4076	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵))
7		df-rex 3067	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
8	2, 6, 7	3bitr4i 292	1 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   ↔ wb 196   ∧ wa 382  ∃wex 1852   ∈ wcel 2145  Ⅎwnfc 2900   ≠ wne 2943  ∃wrex 3062   ∩ cin 3722  ∅c0 4063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064

This theorem is referenced by:  inn0  39765