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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 5311 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | eleq1 2817 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 257 | . . . . 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 7916 | . . . . 5 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V) | |
7 | rnxp 6179 | . . . . . 6 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
8 | 7 | eleq1d 2814 | . . . . 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 3022 | . 2 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (¬ 𝐴 ∈ V → 𝐵 ∈ V)) |
12 | 11 | orrd 861 | 1 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 Vcvv 3473 ∅c0 4326 × cxp 5680 ran crn 5683 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 |
This theorem is referenced by: (None) |
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