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Theorem xpexr 7932
Description: If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
xpexr ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))

Proof of Theorem xpexr
StepHypRef Expression
1 0ex 5311 . . . . . 6 ∅ ∈ V
2 eleq1 2817 . . . . . 6 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 257 . . . . 5 (𝐴 = ∅ → 𝐴 ∈ V)
43pm2.24d 151 . . . 4 (𝐴 = ∅ → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
54a1d 25 . . 3 (𝐴 = ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
6 rnexg 7916 . . . . 5 ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
7 rnxp 6179 . . . . . 6 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
87eleq1d 2814 . . . . 5 (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V))
96, 8imbitrid 243 . . . 4 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶𝐵 ∈ V))
109a1dd 50 . . 3 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
115, 10pm2.61ine 3022 . 2 ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
1211orrd 861 1 ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 845   = wceq 1533  wcel 2098  wne 2937  Vcvv 3473  c0 4326   × cxp 5680  ran crn 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by: (None)
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