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Theorem grur1 10858
Description: A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
gruina.1 𝐴 = (𝑈 ∩ On)
Assertion
Ref Expression
grur1 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1𝐴))

Proof of Theorem grur1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nss 4060 . . . . 5 𝑈 ⊆ (𝑅1𝐴) ↔ ∃𝑥(𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)))
2 fveqeq2 6916 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
32rspcev 3622 . . . . . . . . . 10 ((𝑥𝑈 ∧ (rank‘𝑥) = 𝐴) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴)
43ex 412 . . . . . . . . 9 (𝑥𝑈 → ((rank‘𝑥) = 𝐴 → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
54ad2antrl 728 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ((rank‘𝑥) = 𝐴 → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
6 simplr 769 . . . . . . . . . . . 12 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑈 (𝑅1 “ On))
7 simprl 771 . . . . . . . . . . . 12 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑥𝑈)
8 r1elssi 9843 . . . . . . . . . . . . 13 (𝑈 (𝑅1 “ On) → 𝑈 (𝑅1 “ On))
98sseld 3994 . . . . . . . . . . . 12 (𝑈 (𝑅1 “ On) → (𝑥𝑈𝑥 (𝑅1 “ On)))
106, 7, 9sylc 65 . . . . . . . . . . 11 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑥 (𝑅1 “ On))
11 tcrank 9922 . . . . . . . . . . 11 (𝑥 (𝑅1 “ On) → (rank‘𝑥) = (rank “ (TC‘𝑥)))
1210, 11syl 17 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (rank‘𝑥) = (rank “ (TC‘𝑥)))
1312eleq2d 2825 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank‘𝑥) ↔ 𝐴 ∈ (rank “ (TC‘𝑥))))
14 gruelss 10832 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝑥𝑈)
15 grutr 10831 . . . . . . . . . . . . 13 (𝑈 ∈ Univ → Tr 𝑈)
1615adantr 480 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → Tr 𝑈)
17 vex 3482 . . . . . . . . . . . . 13 𝑥 ∈ V
18 tcmin 9779 . . . . . . . . . . . . 13 (𝑥 ∈ V → ((𝑥𝑈 ∧ Tr 𝑈) → (TC‘𝑥) ⊆ 𝑈))
1917, 18ax-mp 5 . . . . . . . . . . . 12 ((𝑥𝑈 ∧ Tr 𝑈) → (TC‘𝑥) ⊆ 𝑈)
2014, 16, 19syl2anc 584 . . . . . . . . . . 11 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → (TC‘𝑥) ⊆ 𝑈)
21 rankf 9832 . . . . . . . . . . . . 13 rank: (𝑅1 “ On)⟶On
22 ffun 6740 . . . . . . . . . . . . 13 (rank: (𝑅1 “ On)⟶On → Fun rank)
2321, 22ax-mp 5 . . . . . . . . . . . 12 Fun rank
24 fvelima 6974 . . . . . . . . . . . 12 ((Fun rank ∧ 𝐴 ∈ (rank “ (TC‘𝑥))) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
2523, 24mpan 690 . . . . . . . . . . 11 (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
26 ssrexv 4065 . . . . . . . . . . 11 ((TC‘𝑥) ⊆ 𝑈 → (∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴 → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
2720, 25, 26syl2im 40 . . . . . . . . . 10 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
2827ad2ant2r 747 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
2913, 28sylbid 240 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank‘𝑥) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
30 simprr 773 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ¬ 𝑥 ∈ (𝑅1𝐴))
31 ne0i 4347 . . . . . . . . . . . . . . 15 (𝑥𝑈𝑈 ≠ ∅)
32 gruina.1 . . . . . . . . . . . . . . . 16 𝐴 = (𝑈 ∩ On)
3332gruina 10856 . . . . . . . . . . . . . . 15 ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
3431, 33sylan2 593 . . . . . . . . . . . . . 14 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ Inacc)
35 inawina 10728 . . . . . . . . . . . . . 14 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
36 winaon 10726 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw𝐴 ∈ On)
3734, 35, 363syl 18 . . . . . . . . . . . . 13 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ On)
38 r1fnon 9805 . . . . . . . . . . . . . 14 𝑅1 Fn On
39 fndm 6672 . . . . . . . . . . . . . 14 (𝑅1 Fn On → dom 𝑅1 = On)
4038, 39ax-mp 5 . . . . . . . . . . . . 13 dom 𝑅1 = On
4137, 40eleqtrrdi 2850 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ dom 𝑅1)
4241ad2ant2r 747 . . . . . . . . . . 11 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝐴 ∈ dom 𝑅1)
43 rankr1ag 9840 . . . . . . . . . . 11 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
4410, 42, 43syl2anc 584 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
4530, 44mtbid 324 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ¬ (rank‘𝑥) ∈ 𝐴)
46 rankon 9833 . . . . . . . . . . . . 13 (rank‘𝑥) ∈ On
47 eloni 6396 . . . . . . . . . . . . . 14 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
48 eloni 6396 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
49 ordtri3or 6418 . . . . . . . . . . . . . 14 ((Ord (rank‘𝑥) ∧ Ord 𝐴) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
5047, 48, 49syl2an 596 . . . . . . . . . . . . 13 (((rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
5146, 37, 50sylancr 587 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
52 3orass 1089 . . . . . . . . . . . 12 (((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)) ↔ ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5351, 52sylib 218 . . . . . . . . . . 11 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5453ord 864 . . . . . . . . . 10 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5554ad2ant2r 747 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5645, 55mpd 15 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
575, 29, 56mpjaod 860 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴)
5857ex 412 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → ((𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
5958exlimdv 1931 . . . . 5 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (∃𝑥(𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
601, 59biimtrid 242 . . . 4 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1𝐴) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
61 simpll 767 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
62 ne0i 4347 . . . . . . . . . 10 (𝑦𝑈𝑈 ≠ ∅)
6362, 33sylan2 593 . . . . . . . . 9 ((𝑈 ∈ Univ ∧ 𝑦𝑈) → 𝐴 ∈ Inacc)
6463ad2ant2r 747 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ Inacc)
6564, 35, 363syl 18 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ On)
66 simprl 771 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦𝑈)
67 fveq2 6907 . . . . . . . . . 10 ((rank‘𝑦) = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘𝐴))
6867ad2antll 729 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = (cf‘𝐴))
69 elina 10725 . . . . . . . . . . 11 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
7069simp2bi 1145 . . . . . . . . . 10 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
7164, 70syl 17 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘𝐴) = 𝐴)
7268, 71eqtrd 2775 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = 𝐴)
73 rankcf 10815 . . . . . . . . 9 ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
74 fvex 6920 . . . . . . . . . 10 (cf‘(rank‘𝑦)) ∈ V
75 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
76 domtri 10594 . . . . . . . . . 10 (((cf‘(rank‘𝑦)) ∈ V ∧ 𝑦 ∈ V) → ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))))
7774, 75, 76mp2an 692 . . . . . . . . 9 ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦)))
7873, 77mpbir 231 . . . . . . . 8 (cf‘(rank‘𝑦)) ≼ 𝑦
7972, 78eqbrtrrdi 5188 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴𝑦)
80 grudomon 10855 . . . . . . 7 ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦𝑈𝐴𝑦)) → 𝐴𝑈)
8161, 65, 66, 79, 80syl112anc 1373 . . . . . 6 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴𝑈)
82 elin 3979 . . . . . . . . 9 (𝐴 ∈ (𝑈 ∩ On) ↔ (𝐴𝑈𝐴 ∈ On))
8382biimpri 228 . . . . . . . 8 ((𝐴𝑈𝐴 ∈ On) → 𝐴 ∈ (𝑈 ∩ On))
8483, 32eleqtrrdi 2850 . . . . . . 7 ((𝐴𝑈𝐴 ∈ On) → 𝐴𝐴)
85 ordirr 6404 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
8648, 85syl 17 . . . . . . . 8 (𝐴 ∈ On → ¬ 𝐴𝐴)
8786adantl 481 . . . . . . 7 ((𝐴𝑈𝐴 ∈ On) → ¬ 𝐴𝐴)
8884, 87pm2.21dd 195 . . . . . 6 ((𝐴𝑈𝐴 ∈ On) → 𝑈 ⊆ (𝑅1𝐴))
8981, 65, 88syl2anc 584 . . . . 5 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ⊆ (𝑅1𝐴))
9089rexlimdvaa 3154 . . . 4 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (∃𝑦𝑈 (rank‘𝑦) = 𝐴𝑈 ⊆ (𝑅1𝐴)))
9160, 90syld 47 . . 3 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1𝐴) → 𝑈 ⊆ (𝑅1𝐴)))
9291pm2.18d 127 . 2 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 ⊆ (𝑅1𝐴))
9332grur1a 10857 . . 3 (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)
9493adantr 480 . 2 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (𝑅1𝐴) ⊆ 𝑈)
9592, 94eqssd 4013 1 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  Tr wtr 5265  dom cdm 5689  cima 5692  Ord word 6385  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  cdom 8982  csdm 8983  TCctc 9774  𝑅1cr1 9800  rankcrnk 9801  cfccf 9975  Inaccwcwina 10720  Inacccina 10721  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-tc 9775  df-r1 9802  df-rank 9803  df-card 9977  df-cf 9979  df-acn 9980  df-ac 10154  df-wina 10722  df-ina 10723  df-gru 10829
This theorem is referenced by:  grutsk  10860  bj-grur1  37046  grurankcld  44229
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