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Theorem xpexr 7367
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 5013 . . . . . 6 ∅ ∈ V
2 eleq1 2893 . . . . . 6 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 250 . . . . 5 (𝐴 = ∅ → 𝐴 ∈ V)
43pm2.24d 149 . . . 4 (𝐴 = ∅ → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
54a1d 25 . . 3 (𝐴 = ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
6 rnexg 7358 . . . . 5 ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
7 rnxp 5804 . . . . . 6 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
87eleq1d 2890 . . . . 5 (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V))
96, 8syl5ib 236 . . . 4 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶𝐵 ∈ V))
109a1dd 50 . . 3 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
115, 10pm2.61ine 3081 . 2 ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
1211orrd 896 1 ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 880   = wceq 1658  wcel 2166  wne 2998  Vcvv 3413  c0 4143   × cxp 5339  ran crn 5342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-xp 5347  df-rel 5348  df-cnv 5349  df-dm 5351  df-rn 5352
This theorem is referenced by: (None)
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