Theorem xpexr 7869

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 5242	. . . . . 6 ⊢ ∅ ∈ V
2		eleq1 2824	. . . . . 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 7853	. . . . 5 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
7		rnxp 6134	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
8	7	eleq1d 2821	. . . . 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 3015	. 2 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))
12	11	orrd 864	1 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429  ∅c0 4273   × cxp 5629  ran crn 5632

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by: (None)