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Mirrors > Home > MPE Home > Th. List > xpexr | Structured version Visualization version GIF version |
Description: If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
xpexr | ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | eleq1 2676 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 247 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
4 | 3 | pm2.24d 146 | . . . 4 ⊢ (𝐴 = ∅ → (¬ 𝐴 ∈ V → 𝐵 ∈ V)) |
5 | 4 | a1d 25 | . . 3 ⊢ (𝐴 = ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))) |
6 | rnexg 6990 | . . . . 5 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V) | |
7 | rnxp 5483 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
8 | 7 | eleq1d 2672 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V)) |
9 | 6, 8 | syl5ib 233 | . . . 4 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → 𝐵 ∈ V)) |
10 | 9 | a1dd 48 | . . 3 ⊢ (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V))) |
11 | 5, 10 | pm2.61ine 2865 | . 2 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)) |
12 | 11 | orrd 392 | 1 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 × cxp 5036 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: (None) |
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