Theorem bj-xpnzexb 37200

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 37199	. 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V → (𝐴 × 𝐵) ∈ V))
2		eldifsni 4748	. . 3 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → 𝐴 ≠ ∅)
3		bj-xpnzex 37198	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
5	1, 4	impbid 212	1 ⊢ (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900  ∅c0 4287  {csn 4582   × cxp 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643

This theorem is referenced by: (None)