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Theorem gruop 10216
Description: A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
gruop ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Proof of Theorem gruop
StepHypRef Expression
1 dfopg 4774 . . 3 ((𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
213adant1 1127 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 simp1 1133 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
4 grusn 10215 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
543adant3 1129 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴} ∈ 𝑈)
6 grupr 10208 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 grupr 10208 . . 3 ((𝑈 ∈ Univ ∧ {𝐴} ∈ 𝑈 ∧ {𝐴, 𝐵} ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
83, 5, 6, 7syl3anc 1368 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
92, 8eqeltrd 2914 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2114  {csn 4539  {cpr 4541  cop 4545  Univcgru 10201
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
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 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-tr 5149  df-iota 6293  df-fv 6342  df-ov 7143  df-gru 10202
This theorem is referenced by:  gruf  10222
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