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Theorem gruuni 10841
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 5057 . 2 𝐴 = 𝑥𝐴 𝑥
2 gruelss 10835 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
3 dfss3 3971 . . . 4 (𝐴𝑈 ↔ ∀𝑥𝐴 𝑥𝑈)
42, 3sylib 218 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ∀𝑥𝐴 𝑥𝑈)
5 gruiun 10840 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝑥𝑈) → 𝑥𝐴 𝑥𝑈)
64, 5mpd3an3 1463 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝑥𝑈)
71, 6eqeltrid 2844 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  wral 3060  wss 3950   cuni 4906   ciun 4990  Univcgru 10831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-gru 10832
This theorem is referenced by:  gruwun  10854  gruina  10859  grumnudlem  44309
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