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Theorem r1sdom 9192
Description: Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)
Assertion
Ref Expression
r1sdom ((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))

Proof of Theorem r1sdom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2906 . . . 4 (𝑥 = ∅ → (𝐵𝑥𝐵 ∈ ∅))
2 fveq2 6667 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
32breq2d 5075 . . . 4 (𝑥 = ∅ → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘∅)))
41, 3imbi12d 346 . . 3 (𝑥 = ∅ → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))))
5 eleq2 2906 . . . 4 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
6 fveq2 6667 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
76breq2d 5075 . . . 4 (𝑥 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝑦)))
85, 7imbi12d 346 . . 3 (𝑥 = 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦))))
9 eleq2 2906 . . . 4 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ∈ suc 𝑦))
10 fveq2 6667 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110breq2d 5075 . . . 4 (𝑥 = suc 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
129, 11imbi12d 346 . . 3 (𝑥 = suc 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
13 eleq2 2906 . . . 4 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
14 fveq2 6667 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
1514breq2d 5075 . . . 4 (𝑥 = 𝐴 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝐴)))
1613, 15imbi12d 346 . . 3 (𝑥 = 𝐴 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴))))
17 noel 4300 . . . 4 ¬ 𝐵 ∈ ∅
1817pm2.21i 119 . . 3 (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))
19 elsuci 6255 . . . . 5 (𝐵 ∈ suc 𝑦 → (𝐵𝑦𝐵 = 𝑦))
20 sdomtr 8644 . . . . . . . . 9 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))
2120expcom 414 . . . . . . . 8 ((𝑅1𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
22 fvex 6680 . . . . . . . . . 10 (𝑅1𝑦) ∈ V
2322canth2 8659 . . . . . . . . 9 (𝑅1𝑦) ≺ 𝒫 (𝑅1𝑦)
24 r1suc 9188 . . . . . . . . 9 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2523, 24breqtrrid 5101 . . . . . . . 8 (𝑦 ∈ On → (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦))
2621, 25syl11 33 . . . . . . 7 ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑦 ∈ On → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
2726imim2i 16 . . . . . 6 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵𝑦 → (𝑦 ∈ On → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
28 fveq2 6667 . . . . . . . . 9 (𝐵 = 𝑦 → (𝑅1𝐵) = (𝑅1𝑦))
2928breq1d 5073 . . . . . . . 8 (𝐵 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1‘suc 𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)))
3025, 29syl5ibr 247 . . . . . . 7 (𝐵 = 𝑦 → (𝑦 ∈ On → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
3130a1i 11 . . . . . 6 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵 = 𝑦 → (𝑦 ∈ On → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
3227, 31jaod 855 . . . . 5 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → ((𝐵𝑦𝐵 = 𝑦) → (𝑦 ∈ On → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
3319, 32syl5 34 . . . 4 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵 ∈ suc 𝑦 → (𝑦 ∈ On → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
3433com3r 87 . . 3 (𝑦 ∈ On → ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵 ∈ suc 𝑦 → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
35 limuni 6249 . . . . . . 7 (Lim 𝑥𝑥 = 𝑥)
3635eleq2d 2903 . . . . . 6 (Lim 𝑥 → (𝐵𝑥𝐵 𝑥))
37 eluni2 4841 . . . . . 6 (𝐵 𝑥 ↔ ∃𝑦𝑥 𝐵𝑦)
3836, 37syl6bb 288 . . . . 5 (Lim 𝑥 → (𝐵𝑥 ↔ ∃𝑦𝑥 𝐵𝑦))
39 r19.29 3259 . . . . . . 7 ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → ∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦))
40 fvex 6680 . . . . . . . . . 10 (𝑅1𝑥) ∈ V
41 ssiun2 4968 . . . . . . . . . . 11 (𝑦𝑥 → (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦))
42 vex 3503 . . . . . . . . . . . . 13 𝑥 ∈ V
43 r1lim 9190 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4442, 43mpan 686 . . . . . . . . . . . 12 (Lim 𝑥 → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4544sseq2d 4003 . . . . . . . . . . 11 (Lim 𝑥 → ((𝑅1𝑦) ⊆ (𝑅1𝑥) ↔ (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦)))
4641, 45syl5ibr 247 . . . . . . . . . 10 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
47 ssdomg 8544 . . . . . . . . . 10 ((𝑅1𝑥) ∈ V → ((𝑅1𝑦) ⊆ (𝑅1𝑥) → (𝑅1𝑦) ≼ (𝑅1𝑥)))
4840, 46, 47mpsylsyld 69 . . . . . . . . 9 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ≼ (𝑅1𝑥)))
49 id 22 . . . . . . . . . . 11 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)))
5049imp 407 . . . . . . . . . 10 (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑦))
51 sdomdomtr 8639 . . . . . . . . . . 11 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≼ (𝑅1𝑥)) → (𝑅1𝐵) ≺ (𝑅1𝑥))
5251expcom 414 . . . . . . . . . 10 ((𝑅1𝑦) ≼ (𝑅1𝑥) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5350, 52syl5 34 . . . . . . . . 9 ((𝑅1𝑦) ≼ (𝑅1𝑥) → (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5448, 53syl6 35 . . . . . . . 8 (Lim 𝑥 → (𝑦𝑥 → (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5554rexlimdv 3288 . . . . . . 7 (Lim 𝑥 → (∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5639, 55syl5 34 . . . . . 6 (Lim 𝑥 → ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5756expcomd 417 . . . . 5 (Lim 𝑥 → (∃𝑦𝑥 𝐵𝑦 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5838, 57sylbid 241 . . . 4 (Lim 𝑥 → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5958com23 86 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥))))
604, 8, 12, 16, 18, 34, 59tfinds 7562 . 2 (𝐴 ∈ On → (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴)))
6160imp 407 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 843   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  wss 3940  c0 4295  𝒫 cpw 4542   cuni 4837   ciun 4917   class class class wbr 5063  Oncon0 6189  Lim wlim 6190  suc csuc 6191  cfv 6352  cdom 8496  csdm 8497  𝑅1cr1 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-r1 9182
This theorem is referenced by:  r111  9193  smobeth  9997  r1tskina  10193
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