Theorem inteximm 4263

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 3966	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		vex 2818	. . . 4 ⊢ 𝑥 ∈ V
3	2	ssex 4249	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V)
4	1, 3	syl 14	. 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
5	4	exlimiv 1647	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)

Syntax hints:   → wi 4  ∃wex 1541   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3213  ∩ cint 3951

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4230

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226  df-int 3952

This theorem is referenced by:  intexabim  4266  iinexgm  4268  onintonm  4641  elfi2  7261  elfir  7262  fifo  7269