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Theorem grupr 10213
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 10208 . . . . . . 7 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
21ibi 268 . . . . . 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 4669 . . . . . . . . . 10 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
54eleq1d 2902 . . . . . . . . 9 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
65rspccv 3624 . . . . . . . 8 (∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵𝑈 → {𝑥, 𝐵} ∈ 𝑈))
763ad2ant2 1128 . . . . . . 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 3183 . . . . 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 4668 . . . . . 6 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1211eleq1d 2902 . . . . 5 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
1312rspccv 3624 . . . 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 1105 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  𝒫 cpw 4542  {cpr 4566   cuni 4837  Tr wtr 5169  ran crn 5555  (class class class)co 7150  m cmap 8401  Univcgru 10206
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-tr 5170  df-iota 6313  df-fv 6362  df-ov 7153  df-gru 10207
This theorem is referenced by:  grusn  10220  gruop  10221  gruun  10222  gruwun  10229  intgru  10230
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