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Theorem grur1 10861
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 4047 . . . . 5 𝑈 ⊆ (𝑅1𝐴) ↔ ∃𝑥(𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)))
2 fveqeq2 6914 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
32rspcev 3621 . . . . . . . . . 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 768 . . . . . . . . . . . 12 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑈 (𝑅1 “ On))
7 simprl 770 . . . . . . . . . . . 12 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑥𝑈)
8 r1elssi 9846 . . . . . . . . . . . . 13 (𝑈 (𝑅1 “ On) → 𝑈 (𝑅1 “ On))
98sseld 3981 . . . . . . . . . . . 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 9925 . . . . . . . . . . 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 2826 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank‘𝑥) ↔ 𝐴 ∈ (rank “ (TC‘𝑥))))
14 gruelss 10835 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝑥𝑈)
15 grutr 10834 . . . . . . . . . . . . 13 (𝑈 ∈ Univ → Tr 𝑈)
1615adantr 480 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → Tr 𝑈)
17 vex 3483 . . . . . . . . . . . . 13 𝑥 ∈ V
18 tcmin 9782 . . . . . . . . . . . . 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 9835 . . . . . . . . . . . . 13 rank: (𝑅1 “ On)⟶On
22 ffun 6738 . . . . . . . . . . . . 13 (rank: (𝑅1 “ On)⟶On → Fun rank)
2321, 22ax-mp 5 . . . . . . . . . . . 12 Fun rank
24 fvelima 6973 . . . . . . . . . . . 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 4052 . . . . . . . . . . 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 772 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ¬ 𝑥 ∈ (𝑅1𝐴))
31 ne0i 4340 . . . . . . . . . . . . . . 15 (𝑥𝑈𝑈 ≠ ∅)
32 gruina.1 . . . . . . . . . . . . . . . 16 𝐴 = (𝑈 ∩ On)
3332gruina 10859 . . . . . . . . . . . . . . 15 ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
3431, 33sylan2 593 . . . . . . . . . . . . . 14 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ Inacc)
35 inawina 10731 . . . . . . . . . . . . . 14 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
36 winaon 10729 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw𝐴 ∈ On)
3734, 35, 363syl 18 . . . . . . . . . . . . 13 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ On)
38 r1fnon 9808 . . . . . . . . . . . . . 14 𝑅1 Fn On
39 fndm 6670 . . . . . . . . . . . . . 14 (𝑅1 Fn On → dom 𝑅1 = On)
4038, 39ax-mp 5 . . . . . . . . . . . . 13 dom 𝑅1 = On
4137, 40eleqtrrdi 2851 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ dom 𝑅1)
4241ad2ant2r 747 . . . . . . . . . . 11 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝐴 ∈ dom 𝑅1)
43 rankr1ag 9843 . . . . . . . . . . 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 9836 . . . . . . . . . . . . 13 (rank‘𝑥) ∈ On
47 eloni 6393 . . . . . . . . . . . . . 14 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
48 eloni 6393 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
49 ordtri3or 6415 . . . . . . . . . . . . . 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 1932 . . . . 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 766 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
62 ne0i 4340 . . . . . . . . . 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 770 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦𝑈)
67 fveq2 6905 . . . . . . . . . 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 10728 . . . . . . . . . . 11 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
7069simp2bi 1146 . . . . . . . . . 10 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
7164, 70syl 17 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘𝐴) = 𝐴)
7268, 71eqtrd 2776 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = 𝐴)
73 rankcf 10818 . . . . . . . . 9 ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
74 fvex 6918 . . . . . . . . . 10 (cf‘(rank‘𝑦)) ∈ V
75 vex 3483 . . . . . . . . . 10 𝑦 ∈ V
76 domtri 10597 . . . . . . . . . 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 5182 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴𝑦)
80 grudomon 10858 . . . . . . 7 ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦𝑈𝐴𝑦)) → 𝐴𝑈)
8161, 65, 66, 79, 80syl112anc 1375 . . . . . 6 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴𝑈)
82 elin 3966 . . . . . . . . 9 (𝐴 ∈ (𝑈 ∩ On) ↔ (𝐴𝑈𝐴 ∈ On))
8382biimpri 228 . . . . . . . 8 ((𝐴𝑈𝐴 ∈ On) → 𝐴 ∈ (𝑈 ∩ On))
8483, 32eleqtrrdi 2851 . . . . . . 7 ((𝐴𝑈𝐴 ∈ On) → 𝐴𝐴)
85 ordirr 6401 . . . . . . . . 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 3155 . . . 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 10860 . . 3 (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)
9493adantr 480 . 2 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (𝑅1𝐴) ⊆ 𝑈)
9592, 94eqssd 4000 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 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599   cuni 4906   class class class wbr 5142  Tr wtr 5258  dom cdm 5684  cima 5687  Ord word 6382  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  cdom 8984  csdm 8985  TCctc 9777  𝑅1cr1 9803  rankcrnk 9804  cfccf 9978  Inaccwcwina 10723  Inacccina 10724  Univcgru 10831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-tc 9778  df-r1 9805  df-rank 9806  df-card 9980  df-cf 9982  df-acn 9983  df-ac 10157  df-wina 10725  df-ina 10726  df-gru 10832
This theorem is referenced by:  grutsk  10863  bj-grur1  37065  grurankcld  44257
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