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Theorem r1pwss 8812
Description: Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1pwss (𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))

Proof of Theorem r1pwss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1funlim 8794 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 481 . . . . . 6 Lim dom 𝑅1
3 limord 5937 . . . . . 6 (Lim dom 𝑅1 → Ord dom 𝑅1)
42, 3ax-mp 5 . . . . 5 Ord dom 𝑅1
5 ordsson 7146 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
64, 5ax-mp 5 . . . 4 dom 𝑅1 ⊆ On
7 elfvdm 6373 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
86, 7sseldi 3734 . . 3 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ On)
9 onzsl 7203 . . 3 (𝐵 ∈ On ↔ (𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)))
108, 9sylib 208 . 2 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)))
11 noel 4054 . . . . 5 ¬ 𝐴 ∈ ∅
12 fveq2 6344 . . . . . . . 8 (𝐵 = ∅ → (𝑅1𝐵) = (𝑅1‘∅))
13 r10 8796 . . . . . . . 8 (𝑅1‘∅) = ∅
1412, 13syl6eq 2802 . . . . . . 7 (𝐵 = ∅ → (𝑅1𝐵) = ∅)
1514eleq2d 2817 . . . . . 6 (𝐵 = ∅ → (𝐴 ∈ (𝑅1𝐵) ↔ 𝐴 ∈ ∅))
1615biimpcd 239 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ → 𝐴 ∈ ∅))
1711, 16mtoi 190 . . . 4 (𝐴 ∈ (𝑅1𝐵) → ¬ 𝐵 = ∅)
1817pm2.21d 118 . . 3 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
19 simpl 474 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ (𝑅1𝐵))
20 simpr 479 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 = suc 𝑥)
2120fveq2d 6348 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1𝐵) = (𝑅1‘suc 𝑥))
227adantr 472 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 ∈ dom 𝑅1)
2320, 22eqeltrrd 2832 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24 limsuc 7206 . . . . . . . . . . . 12 (Lim dom 𝑅1 → (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1))
252, 24ax-mp 5 . . . . . . . . . . 11 (𝑥 ∈ dom 𝑅1 ↔ suc 𝑥 ∈ dom 𝑅1)
2623, 25sylibr 224 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
27 r1sucg 8797 . . . . . . . . . 10 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2826, 27syl 17 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
2921, 28eqtrd 2786 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1𝐵) = 𝒫 (𝑅1𝑥))
3019, 29eleqtrd 2833 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
31 elpwi 4304 . . . . . . . 8 (𝐴 ∈ 𝒫 (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
32 sspwb 5058 . . . . . . . 8 (𝐴 ⊆ (𝑅1𝑥) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
3331, 32sylib 208 . . . . . . 7 (𝐴 ∈ 𝒫 (𝑅1𝑥) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
3430, 33syl 17 . . . . . 6 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
3534, 29sseqtr4d 3775 . . . . 5 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
3635ex 449 . . . 4 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
3736rexlimdvw 3164 . . 3 (𝐴 ∈ (𝑅1𝐵) → (∃𝑥 ∈ On 𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
38 r1tr 8804 . . . . . 6 Tr (𝑅1𝐵)
39 simpl 474 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝐴 ∈ (𝑅1𝐵))
40 r1limg 8799 . . . . . . . . . . . 12 ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑥𝐵 (𝑅1𝑥))
417, 40sylan 489 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑥𝐵 (𝑅1𝑥))
4239, 41eleqtrd 2833 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝐴 𝑥𝐵 (𝑅1𝑥))
43 eliun 4668 . . . . . . . . . 10 (𝐴 𝑥𝐵 (𝑅1𝑥) ↔ ∃𝑥𝐵 𝐴 ∈ (𝑅1𝑥))
4442, 43sylib 208 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → ∃𝑥𝐵 𝐴 ∈ (𝑅1𝑥))
45 simprl 811 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝑥𝐵)
46 limsuc 7206 . . . . . . . . . . . . 13 (Lim 𝐵 → (𝑥𝐵 ↔ suc 𝑥𝐵))
4746ad2antlr 765 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (𝑥𝐵 ↔ suc 𝑥𝐵))
4845, 47mpbid 222 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc 𝑥𝐵)
49 limsuc 7206 . . . . . . . . . . . 12 (Lim 𝐵 → (suc 𝑥𝐵 ↔ suc suc 𝑥𝐵))
5049ad2antlr 765 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (suc 𝑥𝐵 ↔ suc suc 𝑥𝐵))
5148, 50mpbid 222 . . . . . . . . . 10 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc suc 𝑥𝐵)
52 r1tr 8804 . . . . . . . . . . . . . . 15 Tr (𝑅1𝑥)
53 simprr 813 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐴 ∈ (𝑅1𝑥))
54 trss 4905 . . . . . . . . . . . . . . 15 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
5552, 53, 54mpsyl 68 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐴 ⊆ (𝑅1𝑥))
5655, 32sylib 208 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
577ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐵 ∈ dom 𝑅1)
58 ordtr1 5920 . . . . . . . . . . . . . . . 16 (Ord dom 𝑅1 → ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
594, 58ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
6045, 57, 59syl2anc 696 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝑥 ∈ dom 𝑅1)
6160, 27syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
6256, 61sseqtr4d 3775 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
63 fvex 6354 . . . . . . . . . . . . 13 (𝑅1‘suc 𝑥) ∈ V
6463elpw2 4969 . . . . . . . . . . . 12 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc 𝑥) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
6562, 64sylibr 224 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc 𝑥))
6660, 25sylib 208 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc 𝑥 ∈ dom 𝑅1)
67 r1sucg 8797 . . . . . . . . . . . 12 (suc 𝑥 ∈ dom 𝑅1 → (𝑅1‘suc suc 𝑥) = 𝒫 (𝑅1‘suc 𝑥))
6866, 67syl 17 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (𝑅1‘suc suc 𝑥) = 𝒫 (𝑅1‘suc 𝑥))
6965, 68eleqtrrd 2834 . . . . . . . . . 10 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥))
70 fveq2 6344 . . . . . . . . . . . 12 (𝑦 = suc suc 𝑥 → (𝑅1𝑦) = (𝑅1‘suc suc 𝑥))
7170eleq2d 2817 . . . . . . . . . . 11 (𝑦 = suc suc 𝑥 → (𝒫 𝐴 ∈ (𝑅1𝑦) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)))
7271rspcev 3441 . . . . . . . . . 10 ((suc suc 𝑥𝐵 ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7351, 69, 72syl2anc 696 . . . . . . . . 9 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7444, 73rexlimddv 3165 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
75 eliun 4668 . . . . . . . 8 (𝒫 𝐴 𝑦𝐵 (𝑅1𝑦) ↔ ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7674, 75sylibr 224 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 𝑦𝐵 (𝑅1𝑦))
77 r1limg 8799 . . . . . . . 8 ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑦𝐵 (𝑅1𝑦))
787, 77sylan 489 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑦𝐵 (𝑅1𝑦))
7976, 78eleqtrrd 2834 . . . . . 6 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ∈ (𝑅1𝐵))
80 trss 4905 . . . . . 6 (Tr (𝑅1𝐵) → (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8138, 79, 80mpsyl 68 . . . . 5 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
8281ex 449 . . . 4 (𝐴 ∈ (𝑅1𝐵) → (Lim 𝐵 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8382adantld 484 . . 3 (𝐴 ∈ (𝑅1𝐵) → ((𝐵 ∈ V ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8418, 37, 833jaod 1533 . 2 (𝐴 ∈ (𝑅1𝐵) → ((𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8510, 84mpd 15 1 (𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1071   = wceq 1624  wcel 2131  wrex 3043  Vcvv 3332  wss 3707  c0 4050  𝒫 cpw 4294   ciun 4664  Tr wtr 4896  dom cdm 5258  Ord word 5875  Oncon0 5876  Lim wlim 5877  suc csuc 5878  Fun wfun 6035  cfv 6041  𝑅1cr1 8790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-r1 8792
This theorem is referenced by:  r1sscl  8813
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