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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 5298 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | eleq1 2813 | . . . . . 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 7889 | . . . . 5 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V) | |
7 | rnxp 6160 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
8 | 7 | eleq1d 2810 | . . . . 5 ⊢ (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V)) |
9 | 6, 8 | imbitrid 243 | . . . 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 3017 | . 2 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)) |
12 | 11 | orrd 860 | 1 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∅c0 4315 × cxp 5665 ran crn 5668 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 |
This theorem is referenced by: (None) |
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