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Mirrors > Home > MPE Home > Th. List > ixpexg | Structured version Visualization version GIF version |
Description: The existence of an infinite Cartesian product. 𝑥 is normally a free-variable parameter in 𝐵. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 25-Jan-2015.) |
Ref | Expression |
---|---|
ixpexg | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniixp 8780 | . . . 4 ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | iunexg 7874 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
3 | xpexg 7662 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) → (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ∈ V) | |
4 | 2, 3 | syldan 591 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ∈ V) |
5 | ssexg 5267 | . . . 4 ⊢ ((∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ∈ V) → ∪ X𝑥 ∈ 𝐴 𝐵 ∈ V) | |
6 | 1, 4, 5 | sylancr 587 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → ∪ X𝑥 ∈ 𝐴 𝐵 ∈ V) |
7 | uniexb 7676 | . . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∪ X𝑥 ∈ 𝐴 𝐵 ∈ V) | |
8 | 6, 7 | sylibr 233 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ixpprc 8778 | . . . 4 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) | |
10 | 0ex 5251 | . . . 4 ⊢ ∅ ∈ V | |
11 | 9, 10 | eqeltrdi 2845 | . . 3 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
12 | 11 | adantr 481 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
13 | 8, 12 | pm2.61ian 809 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2105 ∀wral 3061 Vcvv 3441 ⊆ wss 3898 ∅c0 4269 ∪ cuni 4852 ∪ ciun 4941 × cxp 5618 Xcixp 8756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-fv 6487 df-ixp 8757 |
This theorem is referenced by: konigthlem 10425 prdsbasex 17258 isfunc 17676 isnat 17760 natffn 17762 dmdprd 19696 dprdval 19701 elpt 22829 ptbasin2 22835 ptbasfi 22838 ptrest 35889 upixp 36000 hspval 44492 hspmbl 44512 vonioolem2 44564 vonicclem2 44567 |
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