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Theorem r1sdom 9236
 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 2840 . . . 4 (𝑥 = ∅ → (𝐵𝑥𝐵 ∈ ∅))
2 fveq2 6658 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
32breq2d 5044 . . . 4 (𝑥 = ∅ → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘∅)))
41, 3imbi12d 348 . . 3 (𝑥 = ∅ → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))))
5 eleq2 2840 . . . 4 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
6 fveq2 6658 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
76breq2d 5044 . . . 4 (𝑥 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝑦)))
85, 7imbi12d 348 . . 3 (𝑥 = 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦))))
9 eleq2 2840 . . . 4 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ∈ suc 𝑦))
10 fveq2 6658 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110breq2d 5044 . . . 4 (𝑥 = suc 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
129, 11imbi12d 348 . . 3 (𝑥 = suc 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
13 eleq2 2840 . . . 4 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
14 fveq2 6658 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
1514breq2d 5044 . . . 4 (𝑥 = 𝐴 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝐴)))
1613, 15imbi12d 348 . . 3 (𝑥 = 𝐴 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴))))
17 noel 4230 . . . 4 ¬ 𝐵 ∈ ∅
1817pm2.21i 119 . . 3 (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))
19 elsuci 6235 . . . . 5 (𝐵 ∈ suc 𝑦 → (𝐵𝑦𝐵 = 𝑦))
20 sdomtr 8677 . . . . . . . . 9 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))
2120expcom 417 . . . . . . . 8 ((𝑅1𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
22 fvex 6671 . . . . . . . . . 10 (𝑅1𝑦) ∈ V
2322canth2 8692 . . . . . . . . 9 (𝑅1𝑦) ≺ 𝒫 (𝑅1𝑦)
24 r1suc 9232 . . . . . . . . 9 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2523, 24breqtrrid 5070 . . . . . . . 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 6658 . . . . . . . . 9 (𝐵 = 𝑦 → (𝑅1𝐵) = (𝑅1𝑦))
2928breq1d 5042 . . . . . . . 8 (𝐵 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1‘suc 𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)))
3025, 29syl5ibr 249 . . . . . . 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 6229 . . . . . . 7 (Lim 𝑥𝑥 = 𝑥)
3635eleq2d 2837 . . . . . 6 (Lim 𝑥 → (𝐵𝑥𝐵 𝑥))
37 eluni2 4802 . . . . . 6 (𝐵 𝑥 ↔ ∃𝑦𝑥 𝐵𝑦)
3836, 37bitrdi 290 . . . . 5 (Lim 𝑥 → (𝐵𝑥 ↔ ∃𝑦𝑥 𝐵𝑦))
39 r19.29 3181 . . . . . . 7 ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → ∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦))
40 fvex 6671 . . . . . . . . . 10 (𝑅1𝑥) ∈ V
41 ssiun2 4936 . . . . . . . . . . 11 (𝑦𝑥 → (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦))
42 vex 3413 . . . . . . . . . . . . 13 𝑥 ∈ V
43 r1lim 9234 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4442, 43mpan 689 . . . . . . . . . . . 12 (Lim 𝑥 → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4544sseq2d 3924 . . . . . . . . . . 11 (Lim 𝑥 → ((𝑅1𝑦) ⊆ (𝑅1𝑥) ↔ (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦)))
4641, 45syl5ibr 249 . . . . . . . . . 10 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
47 ssdomg 8573 . . . . . . . . . 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 410 . . . . . . . . . 10 (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑦))
51 sdomdomtr 8672 . . . . . . . . . . 11 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≼ (𝑅1𝑥)) → (𝑅1𝐵) ≺ (𝑅1𝑥))
5251expcom 417 . . . . . . . . . 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 3207 . . . . . . 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 420 . . . . 5 (Lim 𝑥 → (∃𝑦𝑥 𝐵𝑦 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5838, 57sylbid 243 . . . 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 7573 . 2 (𝐴 ∈ On → (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴)))
6160imp 410 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494  ∪ cuni 4798  ∪ ciun 4883   class class class wbr 5032  Oncon0 6169  Lim wlim 6170  suc csuc 6171  ‘cfv 6335   ≼ cdom 8525   ≺ csdm 8526  𝑅1cr1 9224 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-r1 9226 This theorem is referenced by:  r111  9237  smobeth  10046  r1tskina  10242
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