Theorem ixp0 6943

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 3517	. . . 4 ⊢ (¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ 𝐵 = ∅)
2	1	rexbii 2540	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnalim 2522	. . 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 6942	. . . 4 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	con3i 637	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
7		notm0 3517	. . 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 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  ∅c0 3496  Xcixp 6910

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-nul 3497  df-ixp 6911

This theorem is referenced by: (None)