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Theorem tcrank 9843
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 9752 . . 3 (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2 onsuc 7794 . . . . 5 (𝑦 ∈ On → suc 𝑦 ∈ On)
3 fveq2 6868 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
43raleqdv 3321 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5 fveq2 6868 . . . . . . . . 9 (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6 fveq2 6868 . . . . . . . . . 10 (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
76imaeq2d 6050 . . . . . . . . 9 (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
85, 7sseq12d 3970 . . . . . . . 8 (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
98cbvralvw 3241 . . . . . . 7 (∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
104, 9bitrdi 289 . . . . . 6 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
11 fveq2 6868 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
1211raleqdv 3321 . . . . . 6 (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13 simpr 488 . . . . . . . . . . . 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 780 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1‘𝑥))
15 simplr 778 . . . . . . . . . . . . . 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 9757 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → (rank‘𝑧) ∈ 𝑥)
17 fveq2 6868 . . . . . . . . . . . . . . . . . 18 (𝑦 = (rank‘𝑧) → (𝑅1‘𝑦) = (𝑅1‘(rank‘𝑧)))
1817raleqdv 3321 . . . . . . . . . . . . . . . . 17 (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
1918rspcv 3578 . . . . . . . . . . . . . . . 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 9755 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → 𝑧 ∈ ∪ (𝑅1 “ On))
22 r1rankidb 9763 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23 ssralv 4006 . . . . . . . . . . . . . . . 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 9785 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ∪ (𝑅1 “ On) → (rank‘𝑧) = ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥})
2827eleq2d 2849 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) ↔ 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
2928biimpd 231 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
30 rankon 9754 . . . . . . . . . . . . . . . . . . . 20 (rank‘𝑧) ∈ On
3130oneli 6462 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32 eleq2w 2847 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
3332ralbidv 3186 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3433onnminsb 7783 . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
39 rexnal 3115 . . . . . . . . . . . . . . 15 (∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤 ↔ ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
4038, 39sylibr 236 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
4140adantl 485 . . . . . . . . . . . . 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 3126 . . . . . . . . . . . . 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 593 . . . . . . . . . . . 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 1151 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢 ∈ 𝑧)
45 tcid 9693 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
4645elv 3460 . . . . . . . . . . . . . . . 16 𝑧 ⊆ (TC‘𝑧)
4746sseli 3933 . . . . . . . . . . . . . . 15 (𝑢 ∈ 𝑧 → 𝑢 ∈ (TC‘𝑧))
48 fveqeq2 6877 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
4948rspcev 3582 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
5049ex 416 . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
5352sseld 3936 . . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1‘𝑥))
55 rankf 9753 . . . . . . . . . . . . . . . . . . 19 rank:∪ (𝑅1 “ On)⟶On
56 ffn 6692 . . . . . . . . . . . . . . . . . . 19 (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . 18 rank Fn ∪ (𝑅1 “ On)
58 r1tr 9735 . . . . . . . . . . . . . . . . . . . 20 Tr (𝑅1‘𝑥)
59 trel 5216 . . . . . . . . . . . . . . . . . . . 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 9755 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝑅1‘𝑥) → 𝑢 ∈ ∪ (𝑅1 “ On))
62 tcwf 9842 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ∈ ∪ (𝑅1 “ On))
63 fvex 6881 . . . . . . . . . . . . . . . . . . . . 21 (TC‘𝑢) ∈ V
6463r1elss 9765 . . . . . . . . . . . . . . . . . . . 20 ((TC‘𝑢) ∈ ∪ (𝑅1 “ On) ↔ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
6562, 64sylib 220 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
6660, 61, 653syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
67 fvelimab 6940 . . . . . . . . . . . . . . . . . 18 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
6857, 66, 67sylancr 596 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
69 vex 3459 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
7069tcel 9699 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ 𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
71 ssrexv 4007 . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7468, 73sylbid 242 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7544, 54, 74syl2anc 593 . . . . . . . . . . . . . . 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 9754 . . . . . . . . . . . . . . . . . . 19 (rank‘𝑢) ∈ On
78 eloni 6357 . . . . . . . . . . . . . . . . . . . 20 ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
79 eloni 6357 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ On → Ord 𝑤)
80 ordtri3or 6379 . . . . . . . . . . . . . . . . . . . 20 ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8178, 79, 80syl2an 605 . . . . . . . . . . . . . . . . . . 19 (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8277, 31, 81sylancr 596 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
83 3orass 1102 . . . . . . . . . . . . . . . . . 18 (((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)) ↔ ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
8482, 83sylib 220 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
8584orcanai 1016 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8685ad2ant2l 756 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
87863adant2 1145 . . . . . . . . . . . . . 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 871 . . . . . . . . . . . . 13 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
8988rexlimdv3a 3168 . . . . . . . . . . . 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 460 . . . . . . . . . 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 9842 . . . . . . . . . . . . 13 (𝑧 ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ∈ ∪ (𝑅1 “ On))
93 r1elssi 9764 . . . . . . . . . . . . . 14 ((TC‘𝑧) ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ⊆ ∪ (𝑅1 “ On))
94 fvelimab 6940 . . . . . . . . . . . . . 14 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9593, 94sylan2 602 . . . . . . . . . . . . 13 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ∈ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9657, 92, 95sylancr 596 . . . . . . . . . . . 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 485 . . . . . . . . . 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 261 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ (rank “ (TC‘𝑧))))
10099ssrdv 3943 . . . . . . . 8 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
101100ralrimiva 3155 . . . . . . 7 ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102101ex 416 . . . . . 6 (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
10310, 12, 102tfis3 7839 . . . . 5 (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
104 fveq2 6868 . . . . . . 7 (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
105 fveq2 6868 . . . . . . . 8 (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
106105imaeq2d 6050 . . . . . . 7 (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
107104, 106sseq12d 3970 . . . . . 6 (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
108107rspccv 3579 . . . . 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 3157 . . 3 (∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
1111, 110sylbi 219 . 2 (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
112 tcvalg 9692 . . . 4 (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
113 r1rankidb 9763 . . . . 5 (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
114 r1tr 9735 . . . . 5 Tr (𝑅1‘(rank‘𝐴))
115 fvex 6881 . . . . . . 7 (𝑅1‘(rank‘𝐴)) ∈ V
116 sseq2 3963 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
117 treq 5215 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
118116, 117anbi12d 641 . . . . . . 7 (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
119115, 118elab 3639 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
120 intss1 4922 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
121119, 120sylbir 237 . . . . 5 ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
122113, 114, 121sylancl 595 . . . 4 (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123112, 122eqsstrd 3971 . . 3 (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
124 imass2 6092 . . . 4 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
125 ffun 6695 . . . . . . . 8 (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
12655, 125ax-mp 5 . . . . . . 7 Fun rank
127 fvelima 6933 . . . . . . 7 ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
128126, 127mpan 700 . . . . . 6 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129 rankr1ai 9757 . . . . . . . 8 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
130 eleq1 2851 . . . . . . . 8 ((rank‘𝑦) = 𝑥 → ((rank‘𝑦) ∈ (rank‘𝐴) ↔ 𝑥 ∈ (rank‘𝐴)))
131129, 130syl5ibcom 247 . . . . . . 7 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → ((rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴)))
132131rexlimiv 3157 . . . . . 6 (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴))
133128, 132syl 17 . . . . 5 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
134133ssriv 3941 . . . 4 (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
135124, 134sstrdi 3949 . . 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 3954 1 (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∨ w3o 1098   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  {cab 2741  ∀wral 3077  ∃wrex 3087  {crab 3415  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4866  ∩ cint 4906  Tr wtr 5208   “ cima 5651  Ord word 6346  Oncon0 6347  suc csuc 6349  Fun wfun 6516   Fn wfn 6517  ⟶wf 6518  ‘cfv 6522  TCctc 9690  𝑅1cr1 9721  rankcrnk 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-inf2 9597
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-tc 9691  df-r1 9723  df-rank 9724
This theorem is referenced by:  hsmexlem5  10388  grur1  10779
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