Theorem inteximm 3991

Description: The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
inteximm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)

Distinct variable group:   𝑥,𝐴

Proof of Theorem inteximm

Step	Hyp	Ref	Expression
1		intss1 3709	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		vex 2623	. . . 4 ⊢ 𝑥 ∈ V
3	2	ssex 3982	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V)
4	1, 3	syl 14	. 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
5	4	exlimiv 1535	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)

Syntax hints:   → wi 4  ∃wex 1427   ∈ wcel 1439  Vcvv 2620   ⊆ wss 3000  ∩ cint 3694

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-int 3695

This theorem is referenced by:  intexabim  3994  iinexgm  3996  onintonm  4347