Theorem inteximm 4212

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 3917	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		vex 2782	. . . 4 ⊢ 𝑥 ∈ V
3	2	ssex 4200	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V)
4	1, 3	syl 14	. 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
5	4	exlimiv 1624	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)

Syntax hints:   → wi 4  ∃wex 1518   ∈ wcel 2180  Vcvv 2779   ⊆ wss 3177  ∩ cint 3902

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-sep 4181

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-int 3903

This theorem is referenced by:  intexabim  4215  iinexgm  4217  onintonm  4586  elfi2  7107  elfir  7108  fifo  7115