Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > gruun | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
gruun | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniprg 4857 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
2 | 1 | 3adant1 1129 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
3 | uniiun 4988 | . . 3 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 | |
4 | 2, 3 | eqtr3di 2793 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥) |
5 | simp1 1135 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ) | |
6 | grupr 10541 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) | |
7 | vex 3434 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 7 | elpr 4585 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
9 | eleq1a 2834 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑈)) | |
10 | eleq1a 2834 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑈 → (𝑥 = 𝐵 → 𝑥 ∈ 𝑈)) | |
11 | 9, 10 | jaao 952 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑥 ∈ 𝑈)) |
12 | 8, 11 | syl5bi 241 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑈)) |
13 | 12 | ralrimiv 3112 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) |
14 | 13 | 3adant1 1129 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) |
15 | gruiun 10543 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) | |
16 | 5, 6, 14, 15 | syl3anc 1370 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) |
17 | 4, 16 | eqeltrd 2839 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∪ cun 3885 {cpr 4564 ∪ cuni 4840 ∪ ciun 4925 Univcgru 10534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-fv 6435 df-ov 7271 df-oprab 7272 df-mpo 7273 df-map 8605 df-gru 10535 |
This theorem is referenced by: gruxp 10551 grusucd 41807 grumnudlem 41862 |
Copyright terms: Public domain | W3C validator |