Theorem inteximm 4197

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 3902	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		vex 2776	. . . 4 ⊢ 𝑥 ∈ V
3	2	ssex 4185	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V)
4	1, 3	syl 14	. 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
5	4	exlimiv 1622	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)

Syntax hints:   → wi 4  ∃wex 1516   ∈ wcel 2177  Vcvv 2773   ⊆ wss 3167  ∩ cint 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4166

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3173  df-ss 3180  df-int 3888

This theorem is referenced by:  intexabim  4200  iinexgm  4202  onintonm  4569  elfi2  7081  elfir  7082  fifo  7089