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Theorem r1tr 8495
Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1tr Tr (𝑅1𝐴)

Proof of Theorem r1tr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1funlim 8485 . . . . . 6 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 476 . . . . 5 Lim dom 𝑅1
3 limord 5683 . . . . 5 (Lim dom 𝑅1 → Ord dom 𝑅1)
4 ordsson 6854 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
52, 3, 4mp2b 10 . . . 4 dom 𝑅1 ⊆ On
65sseli 3559 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
7 fveq2 6084 . . . . . 6 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
8 r10 8487 . . . . . 6 (𝑅1‘∅) = ∅
97, 8syl6eq 2655 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = ∅)
10 treq 4676 . . . . 5 ((𝑅1𝑥) = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
119, 10syl 17 . . . 4 (𝑥 = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
12 fveq2 6084 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
13 treq 4676 . . . . 5 ((𝑅1𝑥) = (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
1412, 13syl 17 . . . 4 (𝑥 = 𝑦 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
15 fveq2 6084 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
16 treq 4676 . . . . 5 ((𝑅1𝑥) = (𝑅1‘suc 𝑦) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
1715, 16syl 17 . . . 4 (𝑥 = suc 𝑦 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
18 fveq2 6084 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
19 treq 4676 . . . . 5 ((𝑅1𝑥) = (𝑅1𝐴) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
2018, 19syl 17 . . . 4 (𝑥 = 𝐴 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
21 tr0 4682 . . . 4 Tr ∅
22 limsuc 6914 . . . . . . . 8 (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
232, 22ax-mp 5 . . . . . . 7 (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
24 simpr 475 . . . . . . . . 9 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1𝑦))
25 pwtr 4838 . . . . . . . . 9 (Tr (𝑅1𝑦) ↔ Tr 𝒫 (𝑅1𝑦))
2624, 25sylib 206 . . . . . . . 8 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr 𝒫 (𝑅1𝑦))
27 r1sucg 8488 . . . . . . . . 9 (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
28 treq 4676 . . . . . . . . 9 ((𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦) → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1𝑦)))
2927, 28syl 17 . . . . . . . 8 (𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1𝑦)))
3026, 29syl5ibrcom 235 . . . . . . 7 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
3123, 30syl5bir 231 . . . . . 6 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
32 ndmfv 6109 . . . . . . . 8 (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33 treq 4676 . . . . . . . 8 ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3432, 33syl 17 . . . . . . 7 (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3521, 34mpbiri 246 . . . . . 6 (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
3631, 35pm2.61d1 169 . . . . 5 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1‘suc 𝑦))
3736ex 448 . . . 4 (𝑦 ∈ On → (Tr (𝑅1𝑦) → Tr (𝑅1‘suc 𝑦)))
38 triun 4684 . . . . . . . 8 (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr 𝑦𝑥 (𝑅1𝑦))
39 r1limg 8490 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4039ancoms 467 . . . . . . . . 9 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
41 treq 4676 . . . . . . . . 9 ((𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4240, 41syl 17 . . . . . . . 8 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4338, 42syl5ibr 234 . . . . . . 7 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr (𝑅1𝑥)))
4443impancom 454 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 Tr (𝑅1𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥)))
45 ndmfv 6109 . . . . . . . 8 𝑥 ∈ dom 𝑅1 → (𝑅1𝑥) = ∅)
4645, 10syl 17 . . . . . . 7 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1𝑥) ↔ Tr ∅))
4721, 46mpbiri 246 . . . . . 6 𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥))
4844, 47pm2.61d1 169 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 Tr (𝑅1𝑦)) → Tr (𝑅1𝑥))
4948ex 448 . . . 4 (Lim 𝑥 → (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr (𝑅1𝑥)))
5011, 14, 17, 20, 21, 37, 49tfinds 6924 . . 3 (𝐴 ∈ On → Tr (𝑅1𝐴))
516, 50syl 17 . 2 (𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
52 ndmfv 6109 . . . 4 𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = ∅)
53 treq 4676 . . . 4 ((𝑅1𝐴) = ∅ → (Tr (𝑅1𝐴) ↔ Tr ∅))
5452, 53syl 17 . . 3 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1𝐴) ↔ Tr ∅))
5521, 54mpbiri 246 . 2 𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
5651, 55pm2.61i 174 1 Tr (𝑅1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wa 382   = wceq 1474  wcel 1975  wral 2891  wss 3535  c0 3869  𝒫 cpw 4103   ciun 4445  Tr wtr 4670  dom cdm 5024  Ord word 5621  Oncon0 5622  Lim wlim 5623  suc csuc 5624  Fun wfun 5780  cfv 5786  𝑅1cr1 8481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-r1 8483
This theorem is referenced by:  r1tr2  8496  r1ordg  8497  r1ord3g  8498  r1ord2  8500  r1sssuc  8502  r1pwss  8503  r1val1  8505  rankwflemb  8512  r1elwf  8515  r1elssi  8524  uniwf  8538  tcrank  8603  ackbij2lem3  8919  r1limwun  9410  tskr1om2  9442
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