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Theorem grur1 9602
 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 3648 . . . . 5 𝑈 ⊆ (𝑅1𝐴) ↔ ∃𝑥(𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)))
2 fveq2 6158 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (rank‘𝑦) = (rank‘𝑥))
32eqeq1d 2623 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
43rspcev 3299 . . . . . . . . . 10 ((𝑥𝑈 ∧ (rank‘𝑥) = 𝐴) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴)
54ex 450 . . . . . . . . 9 (𝑥𝑈 → ((rank‘𝑥) = 𝐴 → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
65ad2antrl 763 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ((rank‘𝑥) = 𝐴 → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
7 simplr 791 . . . . . . . . . . . 12 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑈 (𝑅1 “ On))
8 simprl 793 . . . . . . . . . . . 12 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑥𝑈)
9 r1elssi 8628 . . . . . . . . . . . . 13 (𝑈 (𝑅1 “ On) → 𝑈 (𝑅1 “ On))
109sseld 3587 . . . . . . . . . . . 12 (𝑈 (𝑅1 “ On) → (𝑥𝑈𝑥 (𝑅1 “ On)))
117, 8, 10sylc 65 . . . . . . . . . . 11 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝑥 (𝑅1 “ On))
12 tcrank 8707 . . . . . . . . . . 11 (𝑥 (𝑅1 “ On) → (rank‘𝑥) = (rank “ (TC‘𝑥)))
1311, 12syl 17 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (rank‘𝑥) = (rank “ (TC‘𝑥)))
1413eleq2d 2684 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank‘𝑥) ↔ 𝐴 ∈ (rank “ (TC‘𝑥))))
15 gruelss 9576 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝑥𝑈)
16 grutr 9575 . . . . . . . . . . . . 13 (𝑈 ∈ Univ → Tr 𝑈)
1716adantr 481 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → Tr 𝑈)
18 vex 3193 . . . . . . . . . . . . 13 𝑥 ∈ V
19 tcmin 8577 . . . . . . . . . . . . 13 (𝑥 ∈ V → ((𝑥𝑈 ∧ Tr 𝑈) → (TC‘𝑥) ⊆ 𝑈))
2018, 19ax-mp 5 . . . . . . . . . . . 12 ((𝑥𝑈 ∧ Tr 𝑈) → (TC‘𝑥) ⊆ 𝑈)
2115, 17, 20syl2anc 692 . . . . . . . . . . 11 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → (TC‘𝑥) ⊆ 𝑈)
22 rankf 8617 . . . . . . . . . . . . 13 rank: (𝑅1 “ On)⟶On
23 ffun 6015 . . . . . . . . . . . . 13 (rank: (𝑅1 “ On)⟶On → Fun rank)
2422, 23ax-mp 5 . . . . . . . . . . . 12 Fun rank
25 fvelima 6215 . . . . . . . . . . . 12 ((Fun rank ∧ 𝐴 ∈ (rank “ (TC‘𝑥))) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
2624, 25mpan 705 . . . . . . . . . . 11 (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
27 ssrexv 3652 . . . . . . . . . . 11 ((TC‘𝑥) ⊆ 𝑈 → (∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴 → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
2821, 26, 27syl2im 40 . . . . . . . . . 10 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
2928ad2ant2r 782 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
3014, 29sylbid 230 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝐴 ∈ (rank‘𝑥) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
31 simprr 795 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ¬ 𝑥 ∈ (𝑅1𝐴))
32 ne0i 3903 . . . . . . . . . . . . . . 15 (𝑥𝑈𝑈 ≠ ∅)
33 gruina.1 . . . . . . . . . . . . . . . 16 𝐴 = (𝑈 ∩ On)
3433gruina 9600 . . . . . . . . . . . . . . 15 ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
3532, 34sylan2 491 . . . . . . . . . . . . . 14 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ Inacc)
36 inawina 9472 . . . . . . . . . . . . . 14 (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
37 winaon 9470 . . . . . . . . . . . . . 14 (𝐴 ∈ Inaccw𝐴 ∈ On)
3835, 36, 373syl 18 . . . . . . . . . . . . 13 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ On)
39 r1fnon 8590 . . . . . . . . . . . . . 14 𝑅1 Fn On
40 fndm 5958 . . . . . . . . . . . . . 14 (𝑅1 Fn On → dom 𝑅1 = On)
4139, 40ax-mp 5 . . . . . . . . . . . . 13 dom 𝑅1 = On
4238, 41syl6eleqr 2709 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → 𝐴 ∈ dom 𝑅1)
4342ad2ant2r 782 . . . . . . . . . . 11 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → 𝐴 ∈ dom 𝑅1)
44 rankr1ag 8625 . . . . . . . . . . 11 ((𝑥 (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
4511, 43, 44syl2anc 692 . . . . . . . . . 10 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
4631, 45mtbid 314 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ¬ (rank‘𝑥) ∈ 𝐴)
47 rankon 8618 . . . . . . . . . . . . 13 (rank‘𝑥) ∈ On
48 eloni 5702 . . . . . . . . . . . . . 14 ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
49 eloni 5702 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
50 ordtri3or 5724 . . . . . . . . . . . . . 14 ((Ord (rank‘𝑥) ∧ Ord 𝐴) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
5148, 49, 50syl2an 494 . . . . . . . . . . . . 13 (((rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
5247, 38, 51sylancr 694 . . . . . . . . . . . 12 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
53 3orass 1039 . . . . . . . . . . . 12 (((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)) ↔ ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5452, 53sylib 208 . . . . . . . . . . 11 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5554ord 392 . . . . . . . . . 10 ((𝑈 ∈ Univ ∧ 𝑥𝑈) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5655ad2ant2r 782 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥))))
5746, 56mpd 15 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ((rank‘𝑥) = 𝐴𝐴 ∈ (rank‘𝑥)))
586, 30, 57mpjaod 396 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴))) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴)
5958ex 450 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → ((𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
6059exlimdv 1858 . . . . 5 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (∃𝑥(𝑥𝑈 ∧ ¬ 𝑥 ∈ (𝑅1𝐴)) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
611, 60syl5bi 232 . . . 4 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1𝐴) → ∃𝑦𝑈 (rank‘𝑦) = 𝐴))
62 simpll 789 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
63 ne0i 3903 . . . . . . . . . 10 (𝑦𝑈𝑈 ≠ ∅)
6463, 34sylan2 491 . . . . . . . . 9 ((𝑈 ∈ Univ ∧ 𝑦𝑈) → 𝐴 ∈ Inacc)
6564ad2ant2r 782 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ Inacc)
6665, 36, 373syl 18 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ On)
67 simprl 793 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦𝑈)
68 fveq2 6158 . . . . . . . . . 10 ((rank‘𝑦) = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘𝐴))
6968ad2antll 764 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = (cf‘𝐴))
70 elina 9469 . . . . . . . . . . 11 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
7170simp2bi 1075 . . . . . . . . . 10 (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
7265, 71syl 17 . . . . . . . . 9 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘𝐴) = 𝐴)
7369, 72eqtrd 2655 . . . . . . . 8 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = 𝐴)
74 rankcf 9559 . . . . . . . . 9 ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
75 fvex 6168 . . . . . . . . . 10 (cf‘(rank‘𝑦)) ∈ V
76 vex 3193 . . . . . . . . . 10 𝑦 ∈ V
77 domtri 9338 . . . . . . . . . 10 (((cf‘(rank‘𝑦)) ∈ V ∧ 𝑦 ∈ V) → ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))))
7875, 76, 77mp2an 707 . . . . . . . . 9 ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦)))
7974, 78mpbir 221 . . . . . . . 8 (cf‘(rank‘𝑦)) ≼ 𝑦
8073, 79syl6eqbrr 4663 . . . . . . 7 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴𝑦)
81 grudomon 9599 . . . . . . 7 ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦𝑈𝐴𝑦)) → 𝐴𝑈)
8262, 66, 67, 80, 81syl112anc 1327 . . . . . 6 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴𝑈)
83 elin 3780 . . . . . . . . 9 (𝐴 ∈ (𝑈 ∩ On) ↔ (𝐴𝑈𝐴 ∈ On))
8483biimpri 218 . . . . . . . 8 ((𝐴𝑈𝐴 ∈ On) → 𝐴 ∈ (𝑈 ∩ On))
8584, 33syl6eleqr 2709 . . . . . . 7 ((𝐴𝑈𝐴 ∈ On) → 𝐴𝐴)
86 ordirr 5710 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
8749, 86syl 17 . . . . . . . 8 (𝐴 ∈ On → ¬ 𝐴𝐴)
8887adantl 482 . . . . . . 7 ((𝐴𝑈𝐴 ∈ On) → ¬ 𝐴𝐴)
8985, 88pm2.21dd 186 . . . . . 6 ((𝐴𝑈𝐴 ∈ On) → 𝑈 ⊆ (𝑅1𝐴))
9082, 66, 89syl2anc 692 . . . . 5 (((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) ∧ (𝑦𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ⊆ (𝑅1𝐴))
9190rexlimdvaa 3027 . . . 4 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (∃𝑦𝑈 (rank‘𝑦) = 𝐴𝑈 ⊆ (𝑅1𝐴)))
9261, 91syld 47 . . 3 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1𝐴) → 𝑈 ⊆ (𝑅1𝐴)))
9392pm2.18d 124 . 2 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 ⊆ (𝑅1𝐴))
9433grur1a 9601 . . 3 (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)
9594adantr 481 . 2 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → (𝑅1𝐴) ⊆ 𝑈)
9693, 95eqssd 3605 1 ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1035   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2908  ∃wrex 2909  Vcvv 3190   ∩ cin 3559   ⊆ wss 3560  ∅c0 3897  𝒫 cpw 4136  ∪ cuni 4409   class class class wbr 4623  Tr wtr 4722  dom cdm 5084   “ cima 5087  Ord word 5691  Oncon0 5692  Fun wfun 5851   Fn wfn 5852  ⟶wf 5853  ‘cfv 5857   ≼ cdom 7913   ≺ csdm 7914  TCctc 8572  𝑅1cr1 8585  rankcrnk 8586  cfccf 8723  Inaccwcwina 9464  Inacccina 9465  Univcgru 9572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-ac2 9245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-tc 8573  df-r1 8587  df-rank 8588  df-card 8725  df-cf 8727  df-acn 8728  df-ac 8899  df-wina 9466  df-ina 9467  df-gru 9573 This theorem is referenced by:  grutsk  9604  bj-grur1  32723
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