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Theorem r1tr 9770
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 9760 . . . . . 6 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 486 . . . . 5 Lim dom 𝑅1
3 limord 6424 . . . . 5 (Lim dom 𝑅1 → Ord dom 𝑅1)
4 ordsson 7769 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
52, 3, 4mp2b 10 . . . 4 dom 𝑅1 ⊆ On
65sseli 3978 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
7 fveq2 6891 . . . . . 6 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
8 r10 9762 . . . . . 6 (𝑅1‘∅) = ∅
97, 8eqtrdi 2788 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = ∅)
10 treq 5273 . . . . 5 ((𝑅1𝑥) = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
119, 10syl 17 . . . 4 (𝑥 = ∅ → (Tr (𝑅1𝑥) ↔ Tr ∅))
12 fveq2 6891 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
13 treq 5273 . . . . 5 ((𝑅1𝑥) = (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
1412, 13syl 17 . . . 4 (𝑥 = 𝑦 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝑦)))
15 fveq2 6891 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
16 treq 5273 . . . . 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 6891 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
19 treq 5273 . . . . 5 ((𝑅1𝑥) = (𝑅1𝐴) → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
2018, 19syl 17 . . . 4 (𝑥 = 𝐴 → (Tr (𝑅1𝑥) ↔ Tr (𝑅1𝐴)))
21 tr0 5278 . . . 4 Tr ∅
22 limsuc 7837 . . . . . . . 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 5452 . . . . . . . . 9 (Tr (𝑅1𝑦) ↔ Tr 𝒫 (𝑅1𝑦))
2624, 25sylib 217 . . . . . . . 8 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr 𝒫 (𝑅1𝑦))
27 r1sucg 9763 . . . . . . . . 9 (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
28 treq 5273 . . . . . . . . 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 246 . . . . . . 7 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
3123, 30biimtrrid 242 . . . . . 6 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
32 ndmfv 6926 . . . . . . . 8 (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
33 treq 5273 . . . . . . . 8 ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3432, 33syl 17 . . . . . . 7 (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
3521, 34mpbiri 257 . . . . . 6 (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
3631, 35pm2.61d1 180 . . . . 5 ((𝑦 ∈ On ∧ Tr (𝑅1𝑦)) → Tr (𝑅1‘suc 𝑦))
3736ex 413 . . . 4 (𝑦 ∈ On → (Tr (𝑅1𝑦) → Tr (𝑅1‘suc 𝑦)))
38 triun 5280 . . . . . . . 8 (∀𝑦𝑥 Tr (𝑅1𝑦) → Tr 𝑦𝑥 (𝑅1𝑦))
39 r1limg 9765 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4039ancoms 459 . . . . . . . . 9 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
41 treq 5273 . . . . . . . . 9 ((𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4240, 41syl 17 . . . . . . . 8 ((Lim 𝑥𝑥 ∈ dom 𝑅1) → (Tr (𝑅1𝑥) ↔ Tr 𝑦𝑥 (𝑅1𝑦)))
4338, 42imbitrrid 245 . . . . . . 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 6926 . . . . . . . 8 𝑥 ∈ dom 𝑅1 → (𝑅1𝑥) = ∅)
4645, 10syl 17 . . . . . . 7 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1𝑥) ↔ Tr ∅))
4721, 46mpbiri 257 . . . . . 6 𝑥 ∈ dom 𝑅1 → Tr (𝑅1𝑥))
4844, 47pm2.61d1 180 . . . . 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 7848 . . 3 (𝐴 ∈ On → Tr (𝑅1𝐴))
516, 50syl 17 . 2 (𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
52 ndmfv 6926 . . . 4 𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = ∅)
53 treq 5273 . . . 4 ((𝑅1𝐴) = ∅ → (Tr (𝑅1𝐴) ↔ Tr ∅))
5452, 53syl 17 . . 3 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1𝐴) ↔ Tr ∅))
5521, 54mpbiri 257 . 2 𝐴 ∈ dom 𝑅1 → Tr (𝑅1𝐴))
5651, 55pm2.61i 182 1 Tr (𝑅1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wss 3948  c0 4322  𝒫 cpw 4602   ciun 4997  Tr wtr 5265  dom cdm 5676  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  Fun wfun 6537  cfv 6543  𝑅1cr1 9756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-r1 9758
This theorem is referenced by:  r1tr2  9771  r1ordg  9772  r1ord3g  9773  r1ord2  9775  r1sssuc  9777  r1pwss  9778  r1val1  9780  rankwflemb  9787  r1elwf  9790  r1elssi  9799  uniwf  9813  tcrank  9878  ackbij2lem3  10235  r1limwun  10730  tskr1om2  10762  inagrud  43045
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