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Theorem bj-xpnzexb 33073
 Description: If the first factor of a product is a nonempty set, then the product is a set if and only if the second factor is a set. (Contributed by BJ, 2-Apr-2019.)
Assertion
Ref Expression
bj-xpnzexb (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V))

Proof of Theorem bj-xpnzexb
StepHypRef Expression
1 bj-xpexg2 33072 . 2 (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V → (𝐴 × 𝐵) ∈ V))
2 eldifsni 4353 . . 3 (𝐴 ∈ (𝑉 ∖ {∅}) → 𝐴 ≠ ∅)
3 bj-xpnzex 33071 . . 3 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
42, 3syl 17 . 2 (𝐴 ∈ (𝑉 ∖ {∅}) → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
51, 4impbid 202 1 (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2030   ≠ wne 2823  Vcvv 3231   ∖ cdif 3604  ∅c0 3948  {csn 4210   × cxp 5141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154 This theorem is referenced by: (None)
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