Theorem ixpprc 6709

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		ixpfn 6694	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴)
2		fndm 5307	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
3		vex 2738	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 4886	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	2, 4	eqeltrrdi 2267	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
6	1, 5	syl 14	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V)
7	6	exlimiv 1596	. . 3 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V)
8	7	con3i 632	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
9		notm0 3441	. 2 ⊢ (¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ X𝑥 ∈ 𝐴 𝐵 = ∅)
10	8, 9	sylib 122	1 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1353  ∃wex 1490   ∈ wcel 2146  Vcvv 2735  ∅c0 3420  dom cdm 4620   Fn wfn 5203  Xcixp 6688

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-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ixp 6689

This theorem is referenced by: (None)