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Theorem r1pwss 8530
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 8512 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 477 . . . . . 6 Lim dom 𝑅1
3 limord 5701 . . . . . 6 (Lim dom 𝑅1 → Ord dom 𝑅1)
42, 3ax-mp 5 . . . . 5 Ord dom 𝑅1
5 ordsson 6881 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
64, 5ax-mp 5 . . . 4 dom 𝑅1 ⊆ On
7 elfvdm 6130 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
86, 7sseldi 3566 . . 3 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ On)
9 onzsl 6938 . . 3 (𝐵 ∈ On ↔ (𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)))
108, 9sylib 207 . 2 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ ∨ ∃𝑥 ∈ On 𝐵 = suc 𝑥 ∨ (𝐵 ∈ V ∧ Lim 𝐵)))
11 noel 3878 . . . . 5 ¬ 𝐴 ∈ ∅
12 fveq2 6103 . . . . . . . 8 (𝐵 = ∅ → (𝑅1𝐵) = (𝑅1‘∅))
13 r10 8514 . . . . . . . 8 (𝑅1‘∅) = ∅
1412, 13syl6eq 2660 . . . . . . 7 (𝐵 = ∅ → (𝑅1𝐵) = ∅)
1514eleq2d 2673 . . . . . 6 (𝐵 = ∅ → (𝐴 ∈ (𝑅1𝐵) ↔ 𝐴 ∈ ∅))
1615biimpcd 238 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ → 𝐴 ∈ ∅))
1711, 16mtoi 189 . . . 4 (𝐴 ∈ (𝑅1𝐵) → ¬ 𝐵 = ∅)
1817pm2.21d 117 . . 3 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
19 simpl 472 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ (𝑅1𝐵))
20 simpr 476 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 = suc 𝑥)
2120fveq2d 6107 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1𝐵) = (𝑅1‘suc 𝑥))
227adantr 480 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 ∈ dom 𝑅1)
2320, 22eqeltrrd 2689 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24 limsuc 6941 . . . . . . . . . . . 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 223 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝑥 ∈ dom 𝑅1)
27 r1sucg 8515 . . . . . . . . . 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 2644 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1𝐵) = 𝒫 (𝑅1𝑥))
3019, 29eleqtrd 2690 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
31 elpwi 4117 . . . . . . . 8 (𝐴 ∈ 𝒫 (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
32 sspwb 4844 . . . . . . . 8 (𝐴 ⊆ (𝑅1𝑥) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
3331, 32sylib 207 . . . . . . 7 (𝐴 ∈ 𝒫 (𝑅1𝑥) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
3430, 33syl 17 . . . . . 6 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
3534, 29sseqtr4d 3605 . . . . 5 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
3635ex 449 . . . 4 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
3736rexlimdvw 3016 . . 3 (𝐴 ∈ (𝑅1𝐵) → (∃𝑥 ∈ On 𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
38 r1tr 8522 . . . . . 6 Tr (𝑅1𝐵)
39 simpl 472 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝐴 ∈ (𝑅1𝐵))
40 r1limg 8517 . . . . . . . . . . . 12 ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑥𝐵 (𝑅1𝑥))
417, 40sylan 487 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑥𝐵 (𝑅1𝑥))
4239, 41eleqtrd 2690 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝐴 𝑥𝐵 (𝑅1𝑥))
43 eliun 4460 . . . . . . . . . 10 (𝐴 𝑥𝐵 (𝑅1𝑥) ↔ ∃𝑥𝐵 𝐴 ∈ (𝑅1𝑥))
4442, 43sylib 207 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → ∃𝑥𝐵 𝐴 ∈ (𝑅1𝑥))
45 simprl 790 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝑥𝐵)
46 limsuc 6941 . . . . . . . . . . . . 13 (Lim 𝐵 → (𝑥𝐵 ↔ suc 𝑥𝐵))
4746ad2antlr 759 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (𝑥𝐵 ↔ suc 𝑥𝐵))
4845, 47mpbid 221 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc 𝑥𝐵)
49 limsuc 6941 . . . . . . . . . . . 12 (Lim 𝐵 → (suc 𝑥𝐵 ↔ suc suc 𝑥𝐵))
5049ad2antlr 759 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (suc 𝑥𝐵 ↔ suc suc 𝑥𝐵))
5148, 50mpbid 221 . . . . . . . . . 10 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc suc 𝑥𝐵)
52 r1tr 8522 . . . . . . . . . . . . . . 15 Tr (𝑅1𝑥)
53 simprr 792 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐴 ∈ (𝑅1𝑥))
54 trss 4689 . . . . . . . . . . . . . . 15 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
5552, 53, 54mpsyl 66 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐴 ⊆ (𝑅1𝑥))
5655, 32sylib 207 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1𝑥))
577ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐵 ∈ dom 𝑅1)
58 ordtr1 5684 . . . . . . . . . . . . . . . 16 (Ord dom 𝑅1 → ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
594, 58ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
6045, 57, 59syl2anc 691 . . . . . . . . . . . . . 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 3605 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
63 fvex 6113 . . . . . . . . . . . . 13 (𝑅1‘suc 𝑥) ∈ V
6463elpw2 4755 . . . . . . . . . . . 12 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc 𝑥) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
6562, 64sylibr 223 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc 𝑥))
6660, 25sylib 207 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc 𝑥 ∈ dom 𝑅1)
67 r1sucg 8515 . . . . . . . . . . . 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 2691 . . . . . . . . . 10 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥))
70 fveq2 6103 . . . . . . . . . . . 12 (𝑦 = suc suc 𝑥 → (𝑅1𝑦) = (𝑅1‘suc suc 𝑥))
7170eleq2d 2673 . . . . . . . . . . 11 (𝑦 = suc suc 𝑥 → (𝒫 𝐴 ∈ (𝑅1𝑦) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)))
7271rspcev 3282 . . . . . . . . . 10 ((suc suc 𝑥𝐵 ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7351, 69, 72syl2anc 691 . . . . . . . . 9 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7444, 73rexlimddv 3017 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
75 eliun 4460 . . . . . . . 8 (𝒫 𝐴 𝑦𝐵 (𝑅1𝑦) ↔ ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7674, 75sylibr 223 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 𝑦𝐵 (𝑅1𝑦))
77 r1limg 8517 . . . . . . . 8 ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑦𝐵 (𝑅1𝑦))
787, 77sylan 487 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑦𝐵 (𝑅1𝑦))
7976, 78eleqtrrd 2691 . . . . . 6 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ∈ (𝑅1𝐵))
80 trss 4689 . . . . . 6 (Tr (𝑅1𝐵) → (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8138, 79, 80mpsyl 66 . . . . 5 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
8281ex 449 . . . 4 (𝐴 ∈ (𝑅1𝐵) → (Lim 𝐵 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8382adantld 482 . . 3 (𝐴 ∈ (𝑅1𝐵) → ((𝐵 ∈ V ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8418, 37, 833jaod 1384 . 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 195  wa 383  w3o 1030   = wceq 1475  wcel 1977  wrex 2897  Vcvv 3173  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:  r1sscl  8531
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