Theorem ixpprc 8938

Description: A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain 𝐴, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)

Assertion

Ref	Expression
ixpprc	⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ixpprc

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4332	. . 3 ⊢ (¬ X𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
2		ixpfn 8922	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴)
3		fndm 6646	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4		vex 3468	. . . . . . 7 ⊢ 𝑓 ∈ V
5	4	dmex 7910	. . . . . 6 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2844	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
7	2, 6	syl 17	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V)
8	7	exlimiv 1930	. . 3 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V)
9	1, 8	sylbi 217	. 2 ⊢ (¬ X𝑥 ∈ 𝐴 𝐵 = ∅ → 𝐴 ∈ V)
10	9	con1i 147	1 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3464  ∅c0 4313  dom cdm 5659   Fn wfn 6531  Xcixp 8916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544  df-ixp 8917

This theorem is referenced by:  ixpexg  8941  ixpssmap2g  8946  ixpssmapg  8947  resixpfo  8955