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Theorem gruop 10215
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 4793 . . 3 ((𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
213adant1 1122 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3 simp1 1128 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
4 grusn 10214 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
543adant3 1124 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴} ∈ 𝑈)
6 grupr 10207 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 grupr 10207 . . 3 ((𝑈 ∈ Univ ∧ {𝐴} ∈ 𝑈 ∧ {𝐴, 𝐵} ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
83, 5, 6, 7syl3anc 1363 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
92, 8eqeltrd 2910 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  {csn 4557  {cpr 4559  cop 4563  Univcgru 10200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-tr 5164  df-iota 6307  df-fv 6356  df-ov 7148  df-gru 10201
This theorem is referenced by:  gruf  10221
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