Theorem ixp0 7003

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 3542	. . . 4 ⊢ (¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ 𝐵 = ∅)
2	1	rexbii 2557	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑧 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 = ∅)
3		rexnalim 2539	. . 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 7002	. . . 4 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	con3i 641	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 → ¬ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
7		notm0 3542	. . 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 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∅c0 3520  Xcixp 6970

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-nul 3521  df-ixp 6971

This theorem is referenced by: (None)