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Theorem gruun 9588
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 4546 . . 3 {𝐴, 𝐵} = 𝑥 ∈ {𝐴, 𝐵}𝑥
2 uniprg 4423 . . . 4 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
323adant1 1077 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 3syl5reqr 2670 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) = 𝑥 ∈ {𝐴, 𝐵}𝑥)
5 simp1 1059 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
6 grupr 9579 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 vex 3193 . . . . . . 7 𝑥 ∈ V
87elpr 4176 . . . . . 6 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
9 eleq1a 2693 . . . . . . 7 (𝐴𝑈 → (𝑥 = 𝐴𝑥𝑈))
10 eleq1a 2693 . . . . . . 7 (𝐵𝑈 → (𝑥 = 𝐵𝑥𝑈))
119, 10jaao 531 . . . . . 6 ((𝐴𝑈𝐵𝑈) → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥𝑈))
128, 11syl5bi 232 . . . . 5 ((𝐴𝑈𝐵𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥𝑈))
1312ralrimiv 2961 . . . 4 ((𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
14133adant1 1077 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
15 gruiun 9581 . . 3 ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
165, 6, 14, 15syl3anc 1323 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
174, 16eqeltrd 2698 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  cun 3558  {cpr 4157   cuni 4409   ciun 4492  Univcgru 9572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-gru 9573
This theorem is referenced by:  gruxp  9589
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