Theorem ixpprc 6621

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 6606	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴)
2		fndm 5230	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
3		vex 2692	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 4813	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	2, 4	eqeltrrdi 2232	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
6	1, 5	syl 14	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V)
7	6	exlimiv 1578	. . 3 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V)
8	7	con3i 622	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
9		notm0 3388	. 2 ⊢ (¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ X𝑥 ∈ 𝐴 𝐵 = ∅)
10	8, 9	sylib 121	1 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689  ∅c0 3368  dom cdm 4547   Fn wfn 5126  Xcixp 6600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-ixp 6601

This theorem is referenced by: (None)