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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 8473 | . . . 4 ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | iunexg 7653 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
3 | xpexg 7462 | . . . . 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 5218 | . . . 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 7475 | . . 3 ⊢ (X𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∪ X𝑥 ∈ 𝐴 𝐵 ∈ V) | |
8 | 6, 7 | sylibr 235 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ixpprc 8471 | . . . 4 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) | |
10 | 0ex 5202 | . . . 4 ⊢ ∅ ∈ V | |
11 | 9, 10 | syl6eqel 2918 | . . 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 808 | 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 3135 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 ∪ cuni 4830 ∪ ciun 4910 × cxp 5546 Xcixp 8449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ixp 8450 |
This theorem is referenced by: konigthlem 9978 prdsbasex 16712 isfunc 17122 isnat 17205 natffn 17207 dmdprd 19049 dprdval 19054 elpt 22108 ptbasin2 22114 ptbasfi 22117 ptrest 34772 upixp 34885 hspval 42768 hspmbl 42788 vonioolem2 42840 vonicclem2 42843 |
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