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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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