Theorem xpexr 7941

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

Step	Hyp	Ref	Expression
1		0ex 5313	. . . . . 6 ⊢ ∅ ∈ V
2		eleq1 2827	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
3	1, 2	mpbiri 258	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
4	3	pm2.24d 151	. . . 4 ⊢ (𝐴 = ∅ → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
5	4	a1d 25	. . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
6		rnexg 7925	. . . . 5 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
7		rnxp 6192	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
8	7	eleq1d 2824	. . . . 5 ⊢ (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V))
9	6, 8	imbitrid 244	. . . 4 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → 𝐵 ∈ V))
10	9	a1dd 50	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)))
11	5, 10	pm2.61ine 3023	. 2 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
12	11	orrd 863	1 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478  ∅c0 4339   × cxp 5687  ran crn 5690

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-ext 2706  ax-sep 5302  ax-nul 5312  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-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)