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Theorem r1tr 9051
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 9041 . . . . . 6 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 486 . . . . 5 Lim dom 𝑅1
3 limord 6125 . . . . 5 (Lim dom 𝑅1 → Ord dom 𝑅1)
4 ordsson 7360 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
52, 3, 4mp2b 10 . . . 4 dom 𝑅1 ⊆ On
65sseli 3885 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
7 fveq2 6538 . . . . . 6 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
8 r10 9043 . . . . . 6 (𝑅1‘∅) = ∅
97, 8syl6eq 2847 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = ∅)
10 treq 5069 . . . . 5 ((𝑅1𝑥) = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
119, 10syl 17 . . . 4 (𝑥 = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
12 fveq2 6538 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
13 treq 5069 . . . . 5 ((𝑅1𝑥) = (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
1412, 13syl 17 . . . 4 (𝑥 = 𝑦 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
15 fveq2 6538 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
16 treq 5069 . . . . 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 6538 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
19 treq 5069 . . . . 5 ((𝑅1𝑥) = (𝑅1𝐴) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
2018, 19syl 17 . . . 4 (𝑥 = 𝐴 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
21 tr0 5074 . . . 4 Tr ∅
22 limsuc 7420 . . . . . . . 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 485 . . . . . . . . 9 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1𝑦))
25 pwtr 5237 . . . . . . . . 9 (Tr (𝑅1𝑦) ↔ Tr 𝒫 (𝑅1𝑦))
2624, 25sylib 219 . . . . . . . 8 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr 𝒫 (𝑅1𝑦))
27 r1sucg 9044 . . . . . . . . 9 (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
28 treq 5069 . . . . . . . . 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 248 . . . . . . 7 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
3123, 30syl5bir 244 . . . . . 6 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
32 ndmfv 6568 . . . . . . . 8 (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33 treq 5069 . . . . . . . 8 ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3432, 33syl 17 . . . . . . 7 (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3521, 34mpbiri 259 . . . . . 6 (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
3631, 35pm2.61d1 181 . . . . 5 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1‘suc 𝑦))
3736ex 413 . . . 4 (𝑦 ∈ On → (Tr (𝑅1𝑦) → Tr (𝑅1‘suc 𝑦)))
38 triun 5076 . . . . . . . 8 (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr 𝑦𝑥 (𝑅1𝑦))
39 r1limg 9046 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4039ancoms 459 . . . . . . . . 9 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
41 treq 5069 . . . . . . . . 9 ((𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4240, 41syl 17 . . . . . . . 8 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4338, 42syl5ibr 247 . . . . . . 7 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr (𝑅1𝑥)))
4443impancom 452 . . . . . 6 ((Lim 𝑥 ∧ ∀𝑦𝑥 Tr (𝑅1𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥)))
45 ndmfv 6568 . . . . . . . 8 𝑥 ∈ dom 𝑅1 → (𝑅1𝑥) = ∅)
4645, 10syl 17 . . . . . . 7 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1𝑥) ↔ Tr ∅))
4721, 46mpbiri 259 . . . . . 6 𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥))
4844, 47pm2.61d1 181 . . . . 5 ((Lim 𝑥 ∧ ∀𝑦𝑥 Tr (𝑅1𝑦)) → Tr (𝑅1𝑥))
4948ex 413 . . . 4 (Lim 𝑥 → (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr (𝑅1𝑥)))
5011, 14, 17, 20, 21, 37, 49tfinds 7430 . . 3 (𝐴 ∈ On → Tr (𝑅1𝐴))
516, 50syl 17 . 2 (𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
52 ndmfv 6568 . . . 4 𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = ∅)
53 treq 5069 . . . 4 ((𝑅1𝐴) = ∅ → (Tr (𝑅1𝐴) ↔ Tr ∅))
5452, 53syl 17 . . 3 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1𝐴) ↔ Tr ∅))
5521, 54mpbiri 259 . 2 𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
5651, 55pm2.61i 183 1 Tr (𝑅1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  wss 3859  c0 4211  𝒫 cpw 4453   ciun 4825  Tr wtr 5063  dom cdm 5443  Ord word 6065  Oncon0 6066  Lim wlim 6067  suc csuc 6068  Fun wfun 6219  cfv 6225  𝑅1cr1 9037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-r1 9039
This theorem is referenced by:  r1tr2  9052  r1ordg  9053  r1ord3g  9054  r1ord2  9056  r1sssuc  9058  r1pwss  9059  r1val1  9061  rankwflemb  9068  r1elwf  9071  r1elssi  9080  uniwf  9094  tcrank  9159  ackbij2lem3  9509  r1limwun  10004  tskr1om2  10036  inagrud  40129
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