Theorem ixp0 6721

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 3441	. . . 4 ⊢ (¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ 𝐵 = ∅)
2	1	rexbii 2482	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnalim 2464	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
4	2, 3	sylbir 135	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
5		ixpm 6720	. . . 4 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	con3i 632	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
7		notm0 3441	. . 3 ⊢ (¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ X𝑥 ∈ 𝐴 𝐵 = ∅)
8	6, 7	sylib 122	. 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 1353  ∃wex 1490   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  ∅c0 3420  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-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  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-nul 3421  df-ixp 6689

This theorem is referenced by: (None)