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Theorem gruun 10205
Description: A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruun ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem gruun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uniiun 4955 . . 3 {𝐴, 𝐵} = 𝑥 ∈ {𝐴, 𝐵}𝑥
2 uniprg 4829 . . . 4 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
323adant1 1127 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 3syl5reqr 2871 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) = 𝑥 ∈ {𝐴, 𝐵}𝑥)
5 simp1 1133 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
6 grupr 10196 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 vex 3474 . . . . . . 7 𝑥 ∈ V
87elpr 4563 . . . . . 6 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
9 eleq1a 2907 . . . . . . 7 (𝐴𝑈 → (𝑥 = 𝐴𝑥𝑈))
10 eleq1a 2907 . . . . . . 7 (𝐵𝑈 → (𝑥 = 𝐵𝑥𝑈))
119, 10jaao 952 . . . . . 6 ((𝐴𝑈𝐵𝑈) → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥𝑈))
128, 11syl5bi 245 . . . . 5 ((𝐴𝑈𝐵𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥𝑈))
1312ralrimiv 3169 . . . 4 ((𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
14133adant1 1127 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
15 gruiun 10198 . . 3 ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
165, 6, 14, 15syl3anc 1368 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
174, 16eqeltrd 2912 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wral 3126  cun 3908  {cpr 4542   cuni 4811   ciun 4892  Univcgru 10189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-gru 10190
This theorem is referenced by:  gruxp  10206  grusucd  40721  grumnudlem  40776
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