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Theorem r1pwss 8811
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 8793 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 473 . . . . . 6 Lim dom 𝑅1
3 limord 5927 . . . . . 6 (Lim dom 𝑅1 → Ord dom 𝑅1)
42, 3ax-mp 5 . . . . 5 Ord dom 𝑅1
5 ordsson 7136 . . . . 5 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
64, 5ax-mp 5 . . . 4 dom 𝑅1 ⊆ On
7 elfvdm 6361 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
86, 7sseldi 3750 . . 3 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ On)
9 onzsl 7193 . . 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 4067 . . . . 5 ¬ 𝐴 ∈ ∅
12 fveq2 6332 . . . . . . . 8 (𝐵 = ∅ → (𝑅1𝐵) = (𝑅1‘∅))
13 r10 8795 . . . . . . . 8 (𝑅1‘∅) = ∅
1412, 13syl6eq 2821 . . . . . . 7 (𝐵 = ∅ → (𝑅1𝐵) = ∅)
1514eleq2d 2836 . . . . . 6 (𝐵 = ∅ → (𝐴 ∈ (𝑅1𝐵) ↔ 𝐴 ∈ ∅))
1615biimpcd 239 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ → 𝐴 ∈ ∅))
1711, 16mtoi 190 . . . 4 (𝐴 ∈ (𝑅1𝐵) → ¬ 𝐵 = ∅)
1817pm2.21d 119 . . 3 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = ∅ → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
19 simpl 468 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ (𝑅1𝐵))
20 simpr 471 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 = suc 𝑥)
2120fveq2d 6336 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1𝐵) = (𝑅1‘suc 𝑥))
227adantr 466 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐵 ∈ dom 𝑅1)
2320, 22eqeltrrd 2851 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24 limsuc 7196 . . . . . . . . . . . 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 8796 . . . . . . . . . 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 2805 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → (𝑅1𝐵) = 𝒫 (𝑅1𝑥))
3019, 29eleqtrd 2852 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
31 elpwi 4307 . . . . . . . 8 (𝐴 ∈ 𝒫 (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
32 sspwb 5045 . . . . . . . 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 3791 . . . . 5 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝐵 = suc 𝑥) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
3635ex 397 . . . 4 (𝐴 ∈ (𝑅1𝐵) → (𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
3736rexlimdvw 3182 . . 3 (𝐴 ∈ (𝑅1𝐵) → (∃𝑥 ∈ On 𝐵 = suc 𝑥 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
38 r1tr 8803 . . . . . 6 Tr (𝑅1𝐵)
39 simpl 468 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝐴 ∈ (𝑅1𝐵))
40 r1limg 8798 . . . . . . . . . . . 12 ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑥𝐵 (𝑅1𝑥))
417, 40sylan 569 . . . . . . . . . . 11 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑥𝐵 (𝑅1𝑥))
4239, 41eleqtrd 2852 . . . . . . . . . 10 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝐴 𝑥𝐵 (𝑅1𝑥))
43 eliun 4658 . . . . . . . . . 10 (𝐴 𝑥𝐵 (𝑅1𝑥) ↔ ∃𝑥𝐵 𝐴 ∈ (𝑅1𝑥))
4442, 43sylib 208 . . . . . . . . 9 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → ∃𝑥𝐵 𝐴 ∈ (𝑅1𝑥))
45 simprl 754 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝑥𝐵)
46 limsuc 7196 . . . . . . . . . . . . 13 (Lim 𝐵 → (𝑥𝐵 ↔ suc 𝑥𝐵))
4746ad2antlr 706 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → (𝑥𝐵 ↔ suc 𝑥𝐵))
4845, 47mpbid 222 . . . . . . . . . . 11 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → suc 𝑥𝐵)
49 limsuc 7196 . . . . . . . . . . . 12 (Lim 𝐵 → (suc 𝑥𝐵 ↔ suc suc 𝑥𝐵))
5049ad2antlr 706 . . . . . . . . . . 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 8803 . . . . . . . . . . . . . . 15 Tr (𝑅1𝑥)
53 simprr 756 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐴 ∈ (𝑅1𝑥))
54 trss 4895 . . . . . . . . . . . . . . 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 705 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝐵 ∈ dom 𝑅1)
58 ordtr1 5910 . . . . . . . . . . . . . . . 16 (Ord dom 𝑅1 → ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
594, 58ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
6045, 57, 59syl2anc 573 . . . . . . . . . . . . . 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 3791 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ⊆ (𝑅1‘suc 𝑥))
63 fvex 6342 . . . . . . . . . . . . 13 (𝑅1‘suc 𝑥) ∈ V
6463elpw2 4959 . . . . . . . . . . . 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 8796 . . . . . . . . . . . 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 2853 . . . . . . . . . 10 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥))
70 fveq2 6332 . . . . . . . . . . . 12 (𝑦 = suc suc 𝑥 → (𝑅1𝑦) = (𝑅1‘suc suc 𝑥))
7170eleq2d 2836 . . . . . . . . . . 11 (𝑦 = suc suc 𝑥 → (𝒫 𝐴 ∈ (𝑅1𝑦) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)))
7271rspcev 3460 . . . . . . . . . 10 ((suc suc 𝑥𝐵 ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc 𝑥)) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7351, 69, 72syl2anc 573 . . . . . . . . 9 (((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) ∧ (𝑥𝐵𝐴 ∈ (𝑅1𝑥))) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7444, 73rexlimddv 3183 . . . . . . . 8 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
75 eliun 4658 . . . . . . . 8 (𝒫 𝐴 𝑦𝐵 (𝑅1𝑦) ↔ ∃𝑦𝐵 𝒫 𝐴 ∈ (𝑅1𝑦))
7674, 75sylibr 224 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 𝑦𝐵 (𝑅1𝑦))
77 r1limg 8798 . . . . . . . 8 ((𝐵 ∈ dom 𝑅1 ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑦𝐵 (𝑅1𝑦))
787, 77sylan 569 . . . . . . 7 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → (𝑅1𝐵) = 𝑦𝐵 (𝑅1𝑦))
7976, 78eleqtrrd 2853 . . . . . 6 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ∈ (𝑅1𝐵))
80 trss 4895 . . . . . 6 (Tr (𝑅1𝐵) → (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8138, 79, 80mpsyl 68 . . . . 5 ((𝐴 ∈ (𝑅1𝐵) ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
8281ex 397 . . . 4 (𝐴 ∈ (𝑅1𝐵) → (Lim 𝐵 → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8382adantld 478 . . 3 (𝐴 ∈ (𝑅1𝐵) → ((𝐵 ∈ V ∧ Lim 𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵)))
8418, 37, 833jaod 1540 . 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 382  w3o 1070   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   ciun 4654  Tr wtr 4886  dom cdm 5249  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  Fun wfun 6025  cfv 6031  𝑅1cr1 8789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791
This theorem is referenced by:  r1sscl  8812
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