Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-xpnzexb Structured version   Visualization version   GIF version

Theorem bj-xpnzexb 36944
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 36943 . 2 (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V → (𝐴 × 𝐵) ∈ V))
2 eldifsni 4795 . . 3 (𝐴 ∈ (𝑉 ∖ {∅}) → 𝐴 ≠ ∅)
3 bj-xpnzex 36942 . . 3 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
42, 3syl 17 . 2 (𝐴 ∈ (𝑉 ∖ {∅}) → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
51, 4impbid 212 1 (𝐴 ∈ (𝑉 ∖ {∅}) → (𝐵 ∈ V ↔ (𝐴 × 𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  wne 2938  Vcvv 3478  cdif 3960  c0 4339  {csn 4631   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator