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Theorem r1sdom 8750
 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 2792 . . . 4 (𝑥 = ∅ → (𝐵𝑥𝐵 ∈ ∅))
2 fveq2 6304 . . . . 5 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
32breq2d 4772 . . . 4 (𝑥 = ∅ → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘∅)))
41, 3imbi12d 333 . . 3 (𝑥 = ∅ → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))))
5 eleq2 2792 . . . 4 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
6 fveq2 6304 . . . . 5 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
76breq2d 4772 . . . 4 (𝑥 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝑦)))
85, 7imbi12d 333 . . 3 (𝑥 = 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦))))
9 eleq2 2792 . . . 4 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ∈ suc 𝑦))
10 fveq2 6304 . . . . 5 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110breq2d 4772 . . . 4 (𝑥 = suc 𝑦 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
129, 11imbi12d 333 . . 3 (𝑥 = suc 𝑦 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵 ∈ suc 𝑦 → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))))
13 eleq2 2792 . . . 4 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
14 fveq2 6304 . . . . 5 (𝑥 = 𝐴 → (𝑅1𝑥) = (𝑅1𝐴))
1514breq2d 4772 . . . 4 (𝑥 = 𝐴 → ((𝑅1𝐵) ≺ (𝑅1𝑥) ↔ (𝑅1𝐵) ≺ (𝑅1𝐴)))
1613, 15imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥)) ↔ (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴))))
17 noel 4027 . . . 4 ¬ 𝐵 ∈ ∅
1817pm2.21i 116 . . 3 (𝐵 ∈ ∅ → (𝑅1𝐵) ≺ (𝑅1‘∅))
19 elsuci 5904 . . . . 5 (𝐵 ∈ suc 𝑦 → (𝐵𝑦𝐵 = 𝑦))
20 sdomtr 8214 . . . . . . . . 9 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦))
2120expcom 450 . . . . . . . 8 ((𝑅1𝑦) ≺ (𝑅1‘suc 𝑦) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1‘suc 𝑦)))
22 fvex 6314 . . . . . . . . . 10 (𝑅1𝑦) ∈ V
2322canth2 8229 . . . . . . . . 9 (𝑅1𝑦) ≺ 𝒫 (𝑅1𝑦)
24 r1suc 8746 . . . . . . . . 9 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2523, 24syl5breqr 4798 . . . . . . . 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 6304 . . . . . . . . 9 (𝐵 = 𝑦 → (𝑅1𝐵) = (𝑅1𝑦))
2928breq1d 4770 . . . . . . . 8 (𝐵 = 𝑦 → ((𝑅1𝐵) ≺ (𝑅1‘suc 𝑦) ↔ (𝑅1𝑦) ≺ (𝑅1‘suc 𝑦)))
3025, 29syl5ibr 236 . . . . . . 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 394 . . . . 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 5898 . . . . . . 7 (Lim 𝑥𝑥 = 𝑥)
3635eleq2d 2789 . . . . . 6 (Lim 𝑥 → (𝐵𝑥𝐵 𝑥))
37 eluni2 4548 . . . . . 6 (𝐵 𝑥 ↔ ∃𝑦𝑥 𝐵𝑦)
3836, 37syl6bb 276 . . . . 5 (Lim 𝑥 → (𝐵𝑥 ↔ ∃𝑦𝑥 𝐵𝑦))
39 r19.29 3174 . . . . . . 7 ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → ∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦))
40 fvex 6314 . . . . . . . . . . 11 (𝑅1𝑥) ∈ V
4140a1i 11 . . . . . . . . . 10 (Lim 𝑥 → (𝑅1𝑥) ∈ V)
42 ssiun2 4671 . . . . . . . . . . 11 (𝑦𝑥 → (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦))
43 vex 3307 . . . . . . . . . . . . 13 𝑥 ∈ V
44 r1lim 8748 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4543, 44mpan 708 . . . . . . . . . . . 12 (Lim 𝑥 → (𝑅1𝑥) = 𝑦𝑥 (𝑅1𝑦))
4645sseq2d 3739 . . . . . . . . . . 11 (Lim 𝑥 → ((𝑅1𝑦) ⊆ (𝑅1𝑥) ↔ (𝑅1𝑦) ⊆ 𝑦𝑥 (𝑅1𝑦)))
4742, 46syl5ibr 236 . . . . . . . . . 10 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
48 ssdomg 8118 . . . . . . . . . 10 ((𝑅1𝑥) ∈ V → ((𝑅1𝑦) ⊆ (𝑅1𝑥) → (𝑅1𝑦) ≼ (𝑅1𝑥)))
4941, 47, 48sylsyld 61 . . . . . . . . 9 (Lim 𝑥 → (𝑦𝑥 → (𝑅1𝑦) ≼ (𝑅1𝑥)))
50 id 22 . . . . . . . . . . 11 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)))
5150imp 444 . . . . . . . . . 10 (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑦))
52 sdomdomtr 8209 . . . . . . . . . . 11 (((𝑅1𝐵) ≺ (𝑅1𝑦) ∧ (𝑅1𝑦) ≼ (𝑅1𝑥)) → (𝑅1𝐵) ≺ (𝑅1𝑥))
5352expcom 450 . . . . . . . . . 10 ((𝑅1𝑦) ≼ (𝑅1𝑥) → ((𝑅1𝐵) ≺ (𝑅1𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5451, 53syl5 34 . . . . . . . . 9 ((𝑅1𝑦) ≼ (𝑅1𝑥) → (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5549, 54syl6 35 . . . . . . . 8 (Lim 𝑥 → (𝑦𝑥 → (((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5655rexlimdv 3132 . . . . . . 7 (Lim 𝑥 → (∃𝑦𝑥 ((𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5739, 56syl5 34 . . . . . 6 (Lim 𝑥 → ((∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) ∧ ∃𝑦𝑥 𝐵𝑦) → (𝑅1𝐵) ≺ (𝑅1𝑥)))
5857expcomd 453 . . . . 5 (Lim 𝑥 → (∃𝑦𝑥 𝐵𝑦 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
5938, 58sylbid 230 . . . 4 (Lim 𝑥 → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝑅1𝐵) ≺ (𝑅1𝑥))))
6059com23 86 . . 3 (Lim 𝑥 → (∀𝑦𝑥 (𝐵𝑦 → (𝑅1𝐵) ≺ (𝑅1𝑦)) → (𝐵𝑥 → (𝑅1𝐵) ≺ (𝑅1𝑥))))
614, 8, 12, 16, 18, 34, 60tfinds 7176 . 2 (𝐴 ∈ On → (𝐵𝐴 → (𝑅1𝐵) ≺ (𝑅1𝐴)))
6261imp 444 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1596   ∈ wcel 2103  ∀wral 3014  ∃wrex 3015  Vcvv 3304   ⊆ wss 3680  ∅c0 4023  𝒫 cpw 4266  ∪ cuni 4544  ∪ ciun 4628   class class class wbr 4760  Oncon0 5836  Lim wlim 5837  suc csuc 5838  ‘cfv 6001   ≼ cdom 8070   ≺ csdm 8071  𝑅1cr1 8738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-r1 8740 This theorem is referenced by:  r111  8751  smobeth  9521  r1tskina  9717
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