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Theorem gruuni 9567
Description: A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
Assertion
Ref Expression
gruuni ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)

Proof of Theorem gruuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uniiun 4544 . 2 𝐴 = 𝑥𝐴 𝑥
2 gruelss 9561 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
3 dfss3 3578 . . . 4 (𝐴𝑈 ↔ ∀𝑥𝐴 𝑥𝑈)
42, 3sylib 208 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ∀𝑥𝐴 𝑥𝑈)
5 gruiun 9566 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝑥𝑈) → 𝑥𝐴 𝑥𝑈)
64, 5mpd3an3 1422 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝑥𝑈)
71, 6syl5eqel 2708 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1992  wral 2912  wss 3560   cuni 4407   ciun 4490  Univcgru 9557
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-gru 9558
This theorem is referenced by:  gruwun  9580  gruina  9585
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