Theorem ixp0 6579

Description: The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)

Assertion

Ref	Expression
ixp0	⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Proof of Theorem ixp0

Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		notm0 3349	. . . 4 ⊢ (¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ 𝐵 = ∅)
2	1	rexbii 2416	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnalim 2401	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
4	2, 3	sylbir 134	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
5		ixpm 6578	. . . 4 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	con3i 604	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
7		notm0 3349	. . 3 ⊢ (¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ X𝑥 ∈ 𝐴 𝐵 = ∅)
8	6, 7	sylib 121	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 𝐵 = ∅)
9	4, 8	syl 14	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1314  ∃wex 1451   ∈ wcel 1463  ∀wral 2390  ∃wrex 2391  ∅c0 3329  Xcixp 6546

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-nul 3330  df-ixp 6547

This theorem is referenced by: (None)