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Theorem tcrank 9642
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 9551 . . 3 (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2 suceloni 7659 . . . . 5 (𝑦 ∈ On → suc 𝑦 ∈ On)
3 fveq2 6774 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
43raleqdv 3348 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5 fveq2 6774 . . . . . . . . 9 (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6 fveq2 6774 . . . . . . . . . 10 (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
76imaeq2d 5969 . . . . . . . . 9 (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
85, 7sseq12d 3954 . . . . . . . 8 (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
98cbvralvw 3383 . . . . . . 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 6774 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
1211raleqdv 3348 . . . . . 6 (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13 simpr 485 . . . . . . . . . . . 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 768 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1‘𝑥))
15 simplr 766 . . . . . . . . . . . . . 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 9556 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → (rank‘𝑧) ∈ 𝑥)
17 fveq2 6774 . . . . . . . . . . . . . . . . . 18 (𝑦 = (rank‘𝑧) → (𝑅1‘𝑦) = (𝑅1‘(rank‘𝑧)))
1817raleqdv 3348 . . . . . . . . . . . . . . . . 17 (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
1918rspcv 3557 . . . . . . . . . . . . . . . 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 9554 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1‘𝑥) → 𝑧 ∈ ∪ (𝑅1 “ On))
22 r1rankidb 9562 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23 ssralv 3987 . . . . . . . . . . . . . . . 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 9584 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ∪ (𝑅1 “ On) → (rank‘𝑧) = ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥})
2827eleq2d 2824 . . . . . . . . . . . . . . . . . . 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 9553 . . . . . . . . . . . . . . . . . . . 20 (rank‘𝑧) ∈ On
3130oneli 6374 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32 eleq2w 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
3332ralbidv 3112 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
3433onnminsb 7649 . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
39 rexnal 3169 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 3184 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢 ∈ 𝑧)
45 tcid 9497 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
4645elv 3438 . . . . . . . . . . . . . . . 16 𝑧 ⊆ (TC‘𝑧)
4746sseli 3917 . . . . . . . . . . . . . . 15 (𝑢 ∈ 𝑧 → 𝑢 ∈ (TC‘𝑧))
48 fveqeq2 6783 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
4948rspcev 3561 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
5049ex 413 . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
5352sseld 3920 . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1‘𝑥))
55 rankf 9552 . . . . . . . . . . . . . . . . . . 19 rank:∪ (𝑅1 “ On)⟶On
56 ffn 6600 . . . . . . . . . . . . . . . . . . 19 (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . 18 rank Fn ∪ (𝑅1 “ On)
58 r1tr 9534 . . . . . . . . . . . . . . . . . . . 20 Tr (𝑅1‘𝑥)
59 trel 5198 . . . . . . . . . . . . . . . . . . . 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 9554 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝑅1‘𝑥) → 𝑢 ∈ ∪ (𝑅1 “ On))
62 tcwf 9641 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ∈ ∪ (𝑅1 “ On))
63 fvex 6787 . . . . . . . . . . . . . . . . . . . . 21 (TC‘𝑢) ∈ V
6463r1elss 9564 . . . . . . . . . . . . . . . . . . . 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 6841 . . . . . . . . . . . . . . . . . 18 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
6857, 66, 67sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
69 vex 3436 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
7069tcel 9503 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ 𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
71 ssrexv 3988 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 9553 . . . . . . . . . . . . . . . . . . 19 (rank‘𝑢) ∈ On
78 eloni 6276 . . . . . . . . . . . . . . . . . . . 20 ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
79 eloni 6276 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ On → Ord 𝑤)
80 ordtri3or 6298 . . . . . . . . . . . . . . . . . . . 20 ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8178, 79, 80syl2an 596 . . . . . . . . . . . . . . . . . . 19 (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8277, 31, 81sylancr 587 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
83 3orass 1089 . . . . . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
8685ad2ant2l 743 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
87863adant2 1130 . . . . . . . . . . . . . 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 857 . . . . . . . . . . . . 13 (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
8988rexlimdv3a 3215 . . . . . . . . . . . 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 457 . . . . . . . . . 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 9641 . . . . . . . . . . . . 13 (𝑧 ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ∈ ∪ (𝑅1 “ On))
93 r1elssi 9563 . . . . . . . . . . . . . 14 ((TC‘𝑧) ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ⊆ ∪ (𝑅1 “ On))
94 fvelimab 6841 . . . . . . . . . . . . . 14 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9593, 94sylan2 593 . . . . . . . . . . . . 13 ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ∈ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9657, 92, 95sylancr 587 . . . . . . . . . . . 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 482 . . . . . . . . . 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 258 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ (rank “ (TC‘𝑧))))
10099ssrdv 3927 . . . . . . . 8 (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
101100ralrimiva 3103 . . . . . . 7 ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102101ex 413 . . . . . 6 (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
10310, 12, 102tfis3 7704 . . . . 5 (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
104 fveq2 6774 . . . . . . 7 (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
105 fveq2 6774 . . . . . . . 8 (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
106105imaeq2d 5969 . . . . . . 7 (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
107104, 106sseq12d 3954 . . . . . 6 (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
108107rspccv 3558 . . . . 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 3209 . . 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 9496 . . . 4 (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
113 r1rankidb 9562 . . . . 5 (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
114 r1tr 9534 . . . . 5 Tr (𝑅1‘(rank‘𝐴))
115 fvex 6787 . . . . . . 7 (𝑅1‘(rank‘𝐴)) ∈ V
116 sseq2 3947 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
117 treq 5197 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
118116, 117anbi12d 631 . . . . . . 7 (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
119115, 118elab 3609 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
120 intss1 4894 . . . . . 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 586 . . . 4 (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123112, 122eqsstrd 3959 . . 3 (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
124 imass2 6010 . . . 4 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
125 ffun 6603 . . . . . . . 8 (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
12655, 125ax-mp 5 . . . . . . 7 Fun rank
127 fvelima 6835 . . . . . . 7 ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
128126, 127mpan 687 . . . . . 6 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129 rankr1ai 9556 . . . . . . . 8 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
130 eleq1 2826 . . . . . . . 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 3209 . . . . . 6 (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴))
133128, 132syl 17 . . . . 5 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
134133ssriv 3925 . . . 4 (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
135124, 134sstrdi 3933 . . 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 3938 1 (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ∩ cint 4879  Tr wtr 5191   “ cima 5592  Ord word 6265  Oncon0 6266  suc csuc 6268  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  TCctc 9494  𝑅1cr1 9520  rankcrnk 9521
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-tc 9495  df-r1 9522  df-rank 9523
This theorem is referenced by:  hsmexlem5  10186  grur1  10576
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