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Mirrors > Home > MPE Home > Th. List > gruop | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
Ref | Expression |
---|---|
gruop | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 4793 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | 1 | 3adant1 1122 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) |
3 | simp1 1128 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ) | |
4 | grusn 10214 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) | |
5 | 4 | 3adant3 1124 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴} ∈ 𝑈) |
6 | grupr 10207 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) | |
7 | grupr 10207 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴} ∈ 𝑈 ∧ {𝐴, 𝐵} ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈) | |
8 | 3, 5, 6, 7 | syl3anc 1363 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈) |
9 | 2, 8 | eqeltrd 2910 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 〈𝐴, 𝐵〉 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {csn 4557 {cpr 4559 〈cop 4563 Univcgru 10200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-tr 5164 df-iota 6307 df-fv 6356 df-ov 7148 df-gru 10201 |
This theorem is referenced by: gruf 10221 |
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