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Theorem gruuni 10838
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 5063 . 2 𝐴 = 𝑥𝐴 𝑥
2 gruelss 10832 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
3 dfss3 3984 . . . 4 (𝐴𝑈 ↔ ∀𝑥𝐴 𝑥𝑈)
42, 3sylib 218 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ∀𝑥𝐴 𝑥𝑈)
5 gruiun 10837 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝑥𝑈) → 𝑥𝐴 𝑥𝑈)
64, 5mpd3an3 1461 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝑥𝑈)
71, 6eqeltrid 2843 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  wss 3963   cuni 4912   ciun 4996  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-gru 10829
This theorem is referenced by:  gruwun  10851  gruina  10856  grumnudlem  44281
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