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Theorem grupr 10822
Description: A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grupr ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Proof of Theorem grupr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 10817 . . . . . . 7 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
21ibi 266 . . . . . 6 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)))
32simprd 494 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
4 preq2 4740 . . . . . . . . . 10 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
54eleq1d 2810 . . . . . . . . 9 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
65rspccv 3603 . . . . . . . 8 (∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵𝑈 → {𝑥, 𝐵} ∈ 𝑈))
763ad2ant2 1131 . . . . . . 7 ((𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → (𝐵𝑈 → {𝑥, 𝐵} ∈ 𝑈))
87com12 32 . . . . . 6 (𝐵𝑈 → ((𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → {𝑥, 𝐵} ∈ 𝑈))
98ralimdv 3158 . . . . 5 (𝐵𝑈 → (∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → ∀𝑥𝑈 {𝑥, 𝐵} ∈ 𝑈))
103, 9syl5com 31 . . . 4 (𝑈 ∈ Univ → (𝐵𝑈 → ∀𝑥𝑈 {𝑥, 𝐵} ∈ 𝑈))
11 preq1 4739 . . . . . 6 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1211eleq1d 2810 . . . . 5 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
1312rspccv 3603 . . . 4 (∀𝑥𝑈 {𝑥, 𝐵} ∈ 𝑈 → (𝐴𝑈 → {𝐴, 𝐵} ∈ 𝑈))
1410, 13syl6 35 . . 3 (𝑈 ∈ Univ → (𝐵𝑈 → (𝐴𝑈 → {𝐴, 𝐵} ∈ 𝑈)))
1514com23 86 . 2 (𝑈 ∈ Univ → (𝐴𝑈 → (𝐵𝑈 → {𝐴, 𝐵} ∈ 𝑈)))
16153imp 1108 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  𝒫 cpw 4604  {cpr 4632   cuni 4909  Tr wtr 5266  ran crn 5679  (class class class)co 7419  m cmap 8845  Univcgru 10815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-tr 5267  df-iota 6501  df-fv 6557  df-ov 7422  df-gru 10816
This theorem is referenced by:  grusn  10829  gruop  10830  gruun  10831  gruwun  10838  intgru  10839
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