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Theorem r1tr 8522
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 8512 . . . . . 6 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 477 . . . . 5 Lim dom 𝑅1
3 limord 5701 . . . . 5 (Lim dom 𝑅1 → Ord dom 𝑅1)
4 ordsson 6881 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
52, 3, 4mp2b 10 . . . 4 dom 𝑅1 ⊆ On
65sseli 3564 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
7 fveq2 6103 . . . . . 6 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
8 r10 8514 . . . . . 6 (𝑅1‘∅) = ∅
97, 8syl6eq 2660 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = ∅)
10 treq 4686 . . . . 5 ((𝑅1𝑥) = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
119, 10syl 17 . . . 4 (𝑥 = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
12 fveq2 6103 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
13 treq 4686 . . . . 5 ((𝑅1𝑥) = (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
1412, 13syl 17 . . . 4 (𝑥 = 𝑦 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
15 fveq2 6103 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
16 treq 4686 . . . . 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 6103 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
19 treq 4686 . . . . 5 ((𝑅1𝑥) = (𝑅1𝐴) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
2018, 19syl 17 . . . 4 (𝑥 = 𝐴 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
21 tr0 4692 . . . 4 Tr ∅
22 limsuc 6941 . . . . . . . 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 476 . . . . . . . . 9 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1𝑦))
25 pwtr 4848 . . . . . . . . 9 (Tr (𝑅1𝑦) ↔ Tr 𝒫 (𝑅1𝑦))
2624, 25sylib 207 . . . . . . . 8 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr 𝒫 (𝑅1𝑦))
27 r1sucg 8515 . . . . . . . . 9 (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
28 treq 4686 . . . . . . . . 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 236 . . . . . . 7 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
3123, 30syl5bir 232 . . . . . 6 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
32 ndmfv 6128 . . . . . . . 8 (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33 treq 4686 . . . . . . . 8 ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3432, 33syl 17 . . . . . . 7 (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3521, 34mpbiri 247 . . . . . 6 (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
3631, 35pm2.61d1 170 . . . . 5 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1‘suc 𝑦))
3736ex 449 . . . 4 (𝑦 ∈ On → (Tr (𝑅1𝑦) → Tr (𝑅1‘suc 𝑦)))
38 triun 4694 . . . . . . . 8 (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr 𝑦𝑥 (𝑅1𝑦))
39 r1limg 8517 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4039ancoms 468 . . . . . . . . 9 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
41 treq 4686 . . . . . . . . 9 ((𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4240, 41syl 17 . . . . . . . 8 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4338, 42syl5ibr 235 . . . . . . 7 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr (𝑅1𝑥)))
4443impancom 455 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 Tr (𝑅1𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥)))
45 ndmfv 6128 . . . . . . . 8 𝑥 ∈ dom 𝑅1 → (𝑅1𝑥) = ∅)
4645, 10syl 17 . . . . . . 7 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1𝑥) ↔ Tr ∅))
4721, 46mpbiri 247 . . . . . 6 𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥))
4844, 47pm2.61d1 170 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 Tr (𝑅1𝑦)) → Tr (𝑅1𝑥))
4948ex 449 . . . 4 (Lim 𝑥 → (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr (𝑅1𝑥)))
5011, 14, 17, 20, 21, 37, 49tfinds 6951 . . 3 (𝐴 ∈ On → Tr (𝑅1𝐴))
516, 50syl 17 . 2 (𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
52 ndmfv 6128 . . . 4 𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = ∅)
53 treq 4686 . . . 4 ((𝑅1𝐴) = ∅ → (Tr (𝑅1𝐴) ↔ Tr ∅))
5452, 53syl 17 . . 3 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1𝐴) ↔ Tr ∅))
5521, 54mpbiri 247 . 2 𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
5651, 55pm2.61i 175 1 Tr (𝑅1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540  c0 3874  𝒫 cpw 4108   ciun 4455  Tr wtr 4680  dom cdm 5038  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Fun wfun 5798  cfv 5804  𝑅1cr1 8508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510
This theorem is referenced by:  r1tr2  8523  r1ordg  8524  r1ord3g  8525  r1ord2  8527  r1sssuc  8529  r1pwss  8530  r1val1  8532  rankwflemb  8539  r1elwf  8542  r1elssi  8551  uniwf  8565  tcrank  8630  ackbij2lem3  8946  r1limwun  9437  tskr1om2  9469
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