MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tcrank Structured version   Visualization version   GIF version
Theorem tcrank 9879
Description: This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below 𝐴. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rank‘𝐴), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TC‘𝐴) has a rank below the rank of 𝐴, since intuitively it contains only the members of 𝐴 and the members of those and so on, but nothing "bigger" than 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcrank (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
Proof of Theorem tcrank
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rankwflemb 9788 . . 3 (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2 onsuc 7799 . . . . 5 (𝑦 ∈ On → suc 𝑦 ∈ On)
3 fveq2 6892 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
43raleqdv 3326 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5 fveq2 6892 . . . . . . . . 9 (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6 fveq2 6892 . . . . . . . . . 10 (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
76imaeq2d 6060 . . . . . . . . 9 (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
85, 7sseq12d 4016 . . . . . . . 8 (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
98cbvralvw 3235 . . . . . . 7 (∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
104, 9bitrdi 287 . . . . . 6 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
11 fveq2 6892 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
1211raleqdv 3326 . . . . . 6 (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13 simpr 486 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)))
14 simprl 770 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1‘𝑥))
15 simplr 768 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
16 rankr1ai 9793 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → (rank‘𝑧) ∈ 𝑥)
17 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑦 = (rank‘𝑧) → (𝑅1‘𝑦) = (𝑅1‘(rank‘𝑧)))
1817raleqdv 3326 . . . . . . . . . . . . . . . . 17 (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
1918rspcv 3609 . . . . . . . . . . . . . . . 16 ((rank‘𝑧) ∈ 𝑥 → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2016, 19syl 17 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
21 r1elwf 9791 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → 𝑧 ∈ ∪ (𝑅1 “ On))
22 r1rankidb 9799 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23 ssralv 4051 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ (𝑅1‘(rank‘𝑧)) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2421, 22, 233syl 18 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2520, 24syld 47 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2614, 15, 25sylc 65 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
27 rankval3b 9821 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ∪ (𝑅1 “ On) → (rank‘𝑧) = ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥})
2827eleq2d 2820 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) ↔ 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
2928biimpd 228 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
30 rankon 9790 . . . . . . . . . . . . . . . . . . . 20 (rank‘𝑧) ∈ On
3130oneli 6479 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32 eleq2w 2818 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
3332ralbidv 3178 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3433onnminsb 7787 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ On → (𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3531, 34syl 17 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → (𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3629, 35sylcom 30 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3721, 36syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3837imp 408 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
39 rexnal 3101 . . . . . . . . . . . . . . 15 (∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤 ↔ ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
4038, 39sylibr 233 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
4140adantl 483 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
42 r19.29 3115 . . . . . . . . . . . . 13 ((∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑢 ∈ 𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
4326, 41, 42syl2anc 585 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢 ∈ 𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
44 simp2 1138 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢 ∈ 𝑧)
45 tcid 9734 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
4645elv 3481 . . . . . . . . . . . . . . . 16 𝑧 ⊆ (TC‘𝑧)
4746sseli 3979 . . . . . . . . . . . . . . 15 (𝑢 ∈ 𝑧 → 𝑢 ∈ (TC‘𝑧))
48 fveqeq2 6901 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
4948rspcev 3613 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
5049ex 414 . . . . . . . . . . . . . . 15 (𝑢 ∈ (TC‘𝑧) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
5144, 47, 503syl 18 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
52 simp3l 1202 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
5352sseld 3982 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → 𝑤 ∈ (rank “ (TC‘𝑢))))
54 simp1l 1198 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1‘𝑥))
55 rankf 9789 . . . . . . . . . . . . . . . . . . 19 rank:∪ (𝑅1 “ On)⟶On
56 ffn 6718 . . . . . . . . . . . . . . . . . . 19 (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . 18 rank Fn ∪ (𝑅1 “ On)
58 r1tr 9771 . . . . . . . . . . . . . . . . . . . 20 Tr (𝑅1‘𝑥)
59 trel 5275 . . . . . . . . . . . . . . . . . . . 20 (Tr (𝑅1‘𝑥) → ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → 𝑢 ∈ (𝑅1‘𝑥)))
6058, 59ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → 𝑢 ∈ (𝑅1‘𝑥))
61 r1elwf 9791 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝑅1‘𝑥) → 𝑢 ∈ ∪ (𝑅1 “ On))
62 tcwf 9878 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ∈ ∪ (𝑅1 “ On))
63 fvex 6905 . . . . . . . . . . . . . . . . . . . . 21 (TC‘𝑢) ∈ V
6463r1elss 9801 . . . . . . . . . . . . . . . . . . . 20 ((TC‘𝑢) ∈ ∪ (𝑅1 “ On) ↔ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
6562, 64sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
6660, 61, 653syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
67 fvelimab 6965 . . . . . . . . . . . . . . . . . 18 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
6857, 66, 67sylancr 588 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
69 vex 3479 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
7069tcel 9740 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ 𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
71 ssrexv 4052 . . . . . . . . . . . . . . . . . . 19 ((TC‘𝑢) ⊆ (TC‘𝑧) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7270, 71syl 17 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ 𝑧 → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7372adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7468, 73sylbid 239 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7544, 54, 74syl2anc 585 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7653, 75syld 47 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
77 rankon 9790 . . . . . . . . . . . . . . . . . . 19 (rank‘𝑢) ∈ On
78 eloni 6375 . . . . . . . . . . . . . . . . . . . 20 ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
79 eloni 6375 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ On → Ord 𝑤)
80 ordtri3or 6397 . . . . . . . . . . . . . . . . . . . 20 ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8178, 79, 80syl2an 597 . . . . . . . . . . . . . . . . . . 19 (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8277, 31, 81sylancr 588 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
83 3orass 1091 . . . . . . . . . . . . . . . . . 18 (((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)) ↔ ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
8482, 83sylib 217 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
8584orcanai 1002 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8685ad2ant2l 745 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
87863adant2 1132 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8851, 76, 87mpjaod 859 . . . . . . . . . . . . 13 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
8988rexlimdv3a 3160 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → (∃𝑢 ∈ 𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9013, 43, 89sylc 65 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
9190expr 458 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank‘𝑧) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
92 tcwf 9878 . . . . . . . . . . . . 13 (𝑧 ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ∈ ∪ (𝑅1 “ On))
93 r1elssi 9800 . . . . . . . . . . . . . 14 ((TC‘𝑧) ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ⊆ ∪ (𝑅1 “ On))
94 fvelimab 6965 . . . . . . . . . . . . . 14 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9593, 94sylan2 594 . . . . . . . . . . . . 13 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ∈ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9657, 92, 95sylancr 588 . . . . . . . . . . . 12 (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9721, 96syl 17 . . . . . . . . . . 11 (𝑧 ∈ (𝑅1‘𝑥) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9897adantl 483 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9991, 98sylibrd 259 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ (rank “ (TC‘𝑧))))
10099ssrdv 3989 . . . . . . . 8 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
101100ralrimiva 3147 . . . . . . 7 ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102101ex 414 . . . . . 6 (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
10310, 12, 102tfis3 7847 . . . . 5 (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
104 fveq2 6892 . . . . . . 7 (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
105 fveq2 6892 . . . . . . . 8 (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
106105imaeq2d 6060 . . . . . . 7 (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
107104, 106sseq12d 4016 . . . . . 6 (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
108107rspccv 3610 . . . . 5 (∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
1092, 103, 1083syl 18 . . . 4 (𝑦 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
110109rexlimiv 3149 . . 3 (∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
1111, 110sylbi 216 . 2 (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
112 tcvalg 9733 . . . 4 (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
113 r1rankidb 9799 . . . . 5 (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
114 r1tr 9771 . . . . 5 Tr (𝑅1‘(rank‘𝐴))
115 fvex 6905 . . . . . . 7 (𝑅1‘(rank‘𝐴)) ∈ V
116 sseq2 4009 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
117 treq 5274 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
118116, 117anbi12d 632 . . . . . . 7 (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
119115, 118elab 3669 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
120 intss1 4968 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
121119, 120sylbir 234 . . . . 5 ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
122113, 114, 121sylancl 587 . . . 4 (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123112, 122eqsstrd 4021 . . 3 (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
124 imass2 6102 . . . 4 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
125 ffun 6721 . . . . . . . 8 (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
12655, 125ax-mp 5 . . . . . . 7 Fun rank
127 fvelima 6958 . . . . . . 7 ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
128126, 127mpan 689 . . . . . 6 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129 rankr1ai 9793 . . . . . . . 8 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
130 eleq1 2822 . . . . . . . 8 ((rank‘𝑦) = 𝑥 → ((rank‘𝑦) ∈ (rank‘𝐴) ↔ 𝑥 ∈ (rank‘𝐴)))
131129, 130syl5ibcom 244 . . . . . . 7 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → ((rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴)))
132131rexlimiv 3149 . . . . . 6 (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴))
133128, 132syl 17 . . . . 5 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
134133ssriv 3987 . . . 4 (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
135124, 134sstrdi 3995 . . 3 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
136123, 135syl 17 . 2 (𝐴 ∈ ∪ (𝑅1 “ On) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
137111, 136eqssd 4000 1 (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ⊆ wss 3949  ∪ cuni 4909  ∩ cint 4951  Tr wtr 5266   “ cima 5680  Ord word 6364  Oncon0 6365  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544  TCctc 9731  𝑅1cr1 9757  rankcrnk 9758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-tc 9732  df-r1 9759  df-rank 9760
This theorem is referenced by:  hsmexlem5  10425  grur1  10815
  Copyright terms: Public domain W3C validator