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Theorem grupr 9825
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 9820 . . . . . . 7 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))))
21ibi 256 . . . . . 6 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈)))
32simprd 483 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))
4 preq2 4406 . . . . . . . . . 10 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
54eleq1d 2835 . . . . . . . . 9 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
65rspccv 3457 . . . . . . . 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 3112 . . . . 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 4405 . . . . . 6 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1211eleq1d 2835 . . . . 5 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
1312rspccv 3457 . . . 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 1101 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  𝒫 cpw 4298  {cpr 4319   cuni 4575  Tr wtr 4887  ran crn 5251  (class class class)co 6796  𝑚 cmap 8013  Univcgru 9818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-tr 4888  df-iota 5993  df-fv 6038  df-ov 6799  df-gru 9819
This theorem is referenced by:  grusn  9832  gruop  9833  gruun  9834  gruwun  9841  intgru  9842
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