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Theorem grupr 10553
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 10548 . . . . . . 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 496 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
4 preq2 4670 . . . . . . . . . 10 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
54eleq1d 2823 . . . . . . . . 9 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
65rspccv 3558 . . . . . . . 8 (∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵𝑈 → {𝑥, 𝐵} ∈ 𝑈))
763ad2ant2 1133 . . . . . . 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 3109 . . . . 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 4669 . . . . . 6 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1211eleq1d 2823 . . . . 5 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
1312rspccv 3558 . . . 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 1110 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  𝒫 cpw 4533  {cpr 4563   cuni 4839  Tr wtr 5191  ran crn 5590  (class class class)co 7275  m cmap 8615  Univcgru 10546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-tr 5192  df-iota 6391  df-fv 6441  df-ov 7278  df-gru 10547
This theorem is referenced by:  grusn  10560  gruop  10561  gruun  10562  gruwun  10569  intgru  10570
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