Theorem inteximm 4240

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 3944	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		vex 2804	. . . 4 ⊢ 𝑥 ∈ V
3	2	ssex 4227	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V)
4	1, 3	syl 14	. 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
5	4	exlimiv 1646	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)

Syntax hints:   → wi 4  ∃wex 1540   ∈ wcel 2201  Vcvv 2801   ⊆ wss 3199  ∩ cint 3929

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212  ax-sep 4208

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-int 3930

This theorem is referenced by:  intexabim  4243  iinexgm  4245  onintonm  4617  elfi2  7176  elfir  7177  fifo  7184