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Theorem r1sdom 9769
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 2823 . . . 4 (𝑥 = ∅ → (𝐵𝑥𝐵 ∈ ∅))
2 fveq2 6892 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
32breq2d 5161 . . . 4 (𝑥 = ∅ → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘∅)))
41, 3imbi12d 345 . . 3 (𝑥 = ∅ → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))))
5 eleq2 2823 . . . 4 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
6 fveq2 6892 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
76breq2d 5161 . . . 4 (𝑥 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝑦)))
85, 7imbi12d 345 . . 3 (𝑥 = 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦))))
9 eleq2 2823 . . . 4 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ∈ suc 𝑦))
10 fveq2 6892 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110breq2d 5161 . . . 4 (𝑥 = suc 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
129, 11imbi12d 345 . . 3 (𝑥 = suc 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
13 eleq2 2823 . . . 4 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
14 fveq2 6892 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
1514breq2d 5161 . . . 4 (𝑥 = 𝐴 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝐴)))
1613, 15imbi12d 345 . . 3 (𝑥 = 𝐴 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴))))
17 noel 4331 . . . 4 ¬ 𝐵 ∈ ∅
1817pm2.21i 119 . . 3 (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))
19 elsuci 6432 . . . . 5 (𝐵 ∈ suc 𝑦 → (𝐵𝑦𝐵 = 𝑦))
20 sdomtr 9115 . . . . . . . . 9 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))
2120expcom 415 . . . . . . . 8 ((𝑅1𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
22 fvex 6905 . . . . . . . . . 10 (𝑅1𝑦) ∈ V
2322canth2 9130 . . . . . . . . 9 (𝑅1𝑦) ≺ 𝒫 (𝑅1𝑦)
24 r1suc 9765 . . . . . . . . 9 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2523, 24breqtrrid 5187 . . . . . . . 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 6892 . . . . . . . . 9 (𝐵 = 𝑦 → (𝑅1𝐵) = (𝑅1𝑦))
2928breq1d 5159 . . . . . . . 8 (𝐵 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1‘suc 𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)))
3025, 29imbitrrid 245 . . . . . . 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 858 . . . . 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 6426 . . . . . . 7 (Lim 𝑥𝑥 = 𝑥)
3635eleq2d 2820 . . . . . 6 (Lim 𝑥 → (𝐵𝑥𝐵 𝑥))
37 eluni2 4913 . . . . . 6 (𝐵 𝑥 ↔ ∃𝑦𝑥 𝐵𝑦)
3836, 37bitrdi 287 . . . . 5 (Lim 𝑥 → (𝐵𝑥 ↔ ∃𝑦𝑥 𝐵𝑦))
39 r19.29 3115 . . . . . . 7 ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → ∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦))
40 fvex 6905 . . . . . . . . . 10 (𝑅1𝑥) ∈ V
41 ssiun2 5051 . . . . . . . . . . 11 (𝑦𝑥 → (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦))
42 vex 3479 . . . . . . . . . . . . 13 𝑥 ∈ V
43 r1lim 9767 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4442, 43mpan 689 . . . . . . . . . . . 12 (Lim 𝑥 → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4544sseq2d 4015 . . . . . . . . . . 11 (Lim 𝑥 → ((𝑅1𝑦) ⊆ (𝑅1𝑥) ↔ (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦)))
4641, 45imbitrrid 245 . . . . . . . . . 10 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
47 ssdomg 8996 . . . . . . . . . 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 408 . . . . . . . . . 10 (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑦))
51 sdomdomtr 9110 . . . . . . . . . . 11 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≼ (𝑅1𝑥)) → (𝑅1𝐵) ≺ (𝑅1𝑥))
5251expcom 415 . . . . . . . . . 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 3154 . . . . . . 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 418 . . . . 5 (Lim 𝑥 → (∃𝑦𝑥 𝐵𝑦 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5838, 57sylbid 239 . . . 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 7849 . 2 (𝐴 ∈ On → (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴)))
6160imp 408 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  wss 3949  c0 4323  𝒫 cpw 4603   cuni 4909   ciun 4998   class class class wbr 5149  Oncon0 6365  Lim wlim 6366  suc csuc 6367  cfv 6544  cdom 8937  csdm 8938  𝑅1cr1 9757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-r1 9759
This theorem is referenced by:  r111  9770  smobeth  10581  r1tskina  10777
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