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Mirrors > Home > MPE Home > Th. List > gruen | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
gruen | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 8006 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑦 𝑦:𝐵–1-1-onto→𝐴) | |
2 | f1ofo 6182 | . . . . . . . . 9 ⊢ (𝑦:𝐵–1-1-onto→𝐴 → 𝑦:𝐵–onto→𝐴) | |
3 | simp3l 1109 | . . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝑦:𝐵–onto→𝐴) | |
4 | forn 6156 | . . . . . . . . . . . . 13 ⊢ (𝑦:𝐵–onto→𝐴 → ran 𝑦 = 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → ran 𝑦 = 𝐴) |
6 | fof 6153 | . . . . . . . . . . . . . 14 ⊢ (𝑦:𝐵–onto→𝐴 → 𝑦:𝐵⟶𝐴) | |
7 | fss 6094 | . . . . . . . . . . . . . 14 ⊢ ((𝑦:𝐵⟶𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈) | |
8 | 6, 7 | sylan 487 | . . . . . . . . . . . . 13 ⊢ ((𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈) |
9 | grurn 9661 | . . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑦:𝐵⟶𝑈) → ran 𝑦 ∈ 𝑈) | |
10 | 8, 9 | syl3an3 1401 | . . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → ran 𝑦 ∈ 𝑈) |
11 | 5, 10 | eqeltrrd 2731 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝐴 ∈ 𝑈) |
12 | 11 | 3expia 1286 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → ((𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝐴 ∈ 𝑈)) |
13 | 12 | expd 451 | . . . . . . . . 9 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦:𝐵–onto→𝐴 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))) |
14 | 2, 13 | syl5 34 | . . . . . . . 8 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦:𝐵–1-1-onto→𝐴 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))) |
15 | 14 | exlimdv 1901 | . . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))) |
16 | 15 | com3r 87 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → 𝐴 ∈ 𝑈))) |
17 | 16 | expdimp 452 | . . . . 5 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → (𝐵 ∈ 𝑈 → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → 𝐴 ∈ 𝑈))) |
18 | 1, 17 | syl7bi 245 | . . . 4 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → (𝐵 ∈ 𝑈 → (𝐵 ≈ 𝐴 → 𝐴 ∈ 𝑈))) |
19 | 18 | impd 446 | . . 3 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → ((𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴) → 𝐴 ∈ 𝑈)) |
20 | 19 | ancoms 468 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈) → ((𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴) → 𝐴 ∈ 𝑈)) |
21 | 20 | 3impia 1280 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∃wex 1744 ∈ wcel 2030 ⊆ wss 3607 class class class wbr 4685 ran crn 5144 ⟶wf 5922 –onto→wfo 5924 –1-1-onto→wf1o 5925 ≈ cen 7994 Univcgru 9650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-map 7901 df-en 7998 df-gru 9651 |
This theorem is referenced by: grudomon 9677 gruina 9678 |
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