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Theorem rankr1c 8631
Description: A relationship between the rank function and the cumulative hierarchy of sets function 𝑅1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1c (𝐴 (𝑅1 “ On) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))

Proof of Theorem rankr1c
StepHypRef Expression
1 id 22 . . . 4 (𝐵 = (rank‘𝐴) → 𝐵 = (rank‘𝐴))
2 rankdmr1 8611 . . . 4 (rank‘𝐴) ∈ dom 𝑅1
31, 2syl6eqel 2706 . . 3 (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1)
43a1i 11 . 2 (𝐴 (𝑅1 “ On) → (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1))
5 elfvdm 6179 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝐵) → suc 𝐵 ∈ dom 𝑅1)
6 r1funlim 8576 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
76simpri 478 . . . . . 6 Lim dom 𝑅1
8 limsuc 6999 . . . . . 6 (Lim dom 𝑅1 → (𝐵 ∈ dom 𝑅1 ↔ suc 𝐵 ∈ dom 𝑅1))
97, 8ax-mp 5 . . . . 5 (𝐵 ∈ dom 𝑅1 ↔ suc 𝐵 ∈ dom 𝑅1)
105, 9sylibr 224 . . . 4 (𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐵 ∈ dom 𝑅1)
1110adantl 482 . . 3 ((¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → 𝐵 ∈ dom 𝑅1)
1211a1i 11 . 2 (𝐴 (𝑅1 “ On) → ((¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → 𝐵 ∈ dom 𝑅1))
13 rankr1clem 8630 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
14 rankr1ag 8612 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵))
159, 14sylan2b 492 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵))
16 rankon 8605 . . . . . . 7 (rank‘𝐴) ∈ On
17 limord 5745 . . . . . . . . . 10 (Lim dom 𝑅1 → Ord dom 𝑅1)
187, 17ax-mp 5 . . . . . . . . 9 Ord dom 𝑅1
19 ordelon 5708 . . . . . . . . 9 ((Ord dom 𝑅1𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
2018, 19mpan 705 . . . . . . . 8 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
2120adantl 482 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
22 onsssuc 5774 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵))
2316, 21, 22sylancr 694 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵))
2415, 23bitr4d 271 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))
2513, 24anbi12d 746 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) ↔ (𝐵 ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ⊆ 𝐵)))
26 eqss 3599 . . . 4 (𝐵 = (rank‘𝐴) ↔ (𝐵 ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ⊆ 𝐵))
2725, 26syl6rbbr 279 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))
2827ex 450 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ dom 𝑅1 → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))))
294, 12, 28pm5.21ndd 369 1 (𝐴 (𝑅1 “ On) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3556   cuni 4404  dom cdm 5076  cima 5079  Ord word 5683  Oncon0 5684  Lim wlim 5685  suc csuc 5686  Fun wfun 5843  cfv 5849  𝑅1cr1 8572  rankcrnk 8573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-r1 8574  df-rank 8575
This theorem is referenced by:  rankidn  8632  rankpwi  8633  rankr1g  8642  r1tskina  9551
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