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Theorem r1sdom 9775
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 2821 . . . 4 (𝑥 = ∅ → (𝐵𝑥𝐵 ∈ ∅))
2 fveq2 6891 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
32breq2d 5160 . . . 4 (𝑥 = ∅ → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘∅)))
41, 3imbi12d 344 . . 3 (𝑥 = ∅ → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))))
5 eleq2 2821 . . . 4 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
6 fveq2 6891 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
76breq2d 5160 . . . 4 (𝑥 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝑦)))
85, 7imbi12d 344 . . 3 (𝑥 = 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦))))
9 eleq2 2821 . . . 4 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ∈ suc 𝑦))
10 fveq2 6891 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110breq2d 5160 . . . 4 (𝑥 = suc 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
129, 11imbi12d 344 . . 3 (𝑥 = suc 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
13 eleq2 2821 . . . 4 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
14 fveq2 6891 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
1514breq2d 5160 . . . 4 (𝑥 = 𝐴 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝐴)))
1613, 15imbi12d 344 . . 3 (𝑥 = 𝐴 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴))))
17 noel 4330 . . . 4 ¬ 𝐵 ∈ ∅
1817pm2.21i 119 . . 3 (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))
19 elsuci 6431 . . . . 5 (𝐵 ∈ suc 𝑦 → (𝐵𝑦𝐵 = 𝑦))
20 sdomtr 9121 . . . . . . . . 9 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))
2120expcom 413 . . . . . . . 8 ((𝑅1𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
22 fvex 6904 . . . . . . . . . 10 (𝑅1𝑦) ∈ V
2322canth2 9136 . . . . . . . . 9 (𝑅1𝑦) ≺ 𝒫 (𝑅1𝑦)
24 r1suc 9771 . . . . . . . . 9 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2523, 24breqtrrid 5186 . . . . . . . 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 6891 . . . . . . . . 9 (𝐵 = 𝑦 → (𝑅1𝐵) = (𝑅1𝑦))
2928breq1d 5158 . . . . . . . 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 856 . . . . 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 6425 . . . . . . 7 (Lim 𝑥𝑥 = 𝑥)
3635eleq2d 2818 . . . . . 6 (Lim 𝑥 → (𝐵𝑥𝐵 𝑥))
37 eluni2 4912 . . . . . 6 (𝐵 𝑥 ↔ ∃𝑦𝑥 𝐵𝑦)
3836, 37bitrdi 287 . . . . 5 (Lim 𝑥 → (𝐵𝑥 ↔ ∃𝑦𝑥 𝐵𝑦))
39 r19.29 3113 . . . . . . 7 ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → ∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦))
40 fvex 6904 . . . . . . . . . 10 (𝑅1𝑥) ∈ V
41 ssiun2 5050 . . . . . . . . . . 11 (𝑦𝑥 → (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦))
42 vex 3477 . . . . . . . . . . . . 13 𝑥 ∈ V
43 r1lim 9773 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4442, 43mpan 687 . . . . . . . . . . . 12 (Lim 𝑥 → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4544sseq2d 4014 . . . . . . . . . . 11 (Lim 𝑥 → ((𝑅1𝑦) ⊆ (𝑅1𝑥) ↔ (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦)))
4641, 45imbitrrid 245 . . . . . . . . . 10 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
47 ssdomg 9002 . . . . . . . . . 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 406 . . . . . . . . . 10 (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑦))
51 sdomdomtr 9116 . . . . . . . . . . 11 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≼ (𝑅1𝑥)) → (𝑅1𝐵) ≺ (𝑅1𝑥))
5251expcom 413 . . . . . . . . . 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 3152 . . . . . . 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 416 . . . . 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 7853 . 2 (𝐴 ∈ On → (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴)))
6160imp 406 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 844   = wceq 1540  wcel 2105  wral 3060  wrex 3069  Vcvv 3473  wss 3948  c0 4322  𝒫 cpw 4602   cuni 4908   ciun 4997   class class class wbr 5148  Oncon0 6364  Lim wlim 6365  suc csuc 6366  cfv 6543  cdom 8943  csdm 8944  𝑅1cr1 9763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-r1 9765
This theorem is referenced by:  r111  9776  smobeth  10587  r1tskina  10783
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