Theorem gruop 10688

Description: A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
gruop	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Proof of Theorem gruop

Step	Hyp	Ref	Expression
1		dfopg 4821	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	3adant1 1130	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
3		simp1 1136	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
4		grusn 10687	. . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)
5	4	3adant3 1132	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴} ∈ 𝑈)
6		grupr 10680	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7		grupr 10680	. . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴} ∈ 𝑈 ∧ {𝐴, 𝐵} ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
8	3, 5, 6, 7	syl3anc 1373	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
9	2, 8	eqeltrd 2829	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  {csn 4574  {cpr 4576  ⟨cop 4580  Univcgru 10673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-tr 5197  df-iota 6433  df-fv 6485  df-ov 7344  df-gru 10674

This theorem is referenced by:  gruf  10694