Theorem bj-xpnzexb 35053

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

Step	Hyp	Ref	Expression
1		bj-xpexg2 35052	. 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V → (𝐴 × 𝐵) ∈ V))
2		eldifsni 4720	. . 3 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → 𝐴 ≠ ∅)
3		bj-xpnzex 35051	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
5	1, 4	impbid 215	1 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2112   ≠ wne 2943  Vcvv 3423   ∖ cdif 3881  ∅c0 4254  {csn 4558   × cxp 5577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pow 5282  ax-pr 5346  ax-un 7563

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5585  df-rel 5586  df-cnv 5587  df-dm 5589  df-rn 5590

This theorem is referenced by: (None)