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Theorem rankr1c 8848
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 8828 . . . 4 (rank‘𝐴) ∈ dom 𝑅1
31, 2syl6eqel 2858 . . 3 (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1)
43a1i 11 . 2 (𝐴 (𝑅1 “ On) → (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1))
5 elfvdm 6361 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝐵) → suc 𝐵 ∈ dom 𝑅1)
6 r1funlim 8793 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
76simpri 473 . . . . . 6 Lim dom 𝑅1
8 limsuc 7196 . . . . . 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 467 . . 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 8847 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
14 rankr1ag 8829 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵))
159, 14sylan2b 573 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵))
16 rankon 8822 . . . . . . 7 (rank‘𝐴) ∈ On
17 limord 5927 . . . . . . . . . 10 (Lim dom 𝑅1 → Ord dom 𝑅1)
187, 17ax-mp 5 . . . . . . . . 9 Ord dom 𝑅1
19 ordelon 5890 . . . . . . . . 9 ((Ord dom 𝑅1𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
2018, 19mpan 662 . . . . . . . 8 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
2120adantl 467 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
22 onsssuc 5956 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵))
2316, 21, 22sylancr 567 . . . . . 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 608 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) ↔ (𝐵 ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ⊆ 𝐵)))
26 eqss 3767 . . . 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 397 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ dom 𝑅1 → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))))
294, 12, 28pm5.21ndd 368 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 382   = wceq 1631  wcel 2145  wss 3723   cuni 4574  dom cdm 5249  cima 5252  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  Fun wfun 6025  cfv 6031  𝑅1cr1 8789  rankcrnk 8790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791  df-rank 8792
This theorem is referenced by:  rankidn  8849  rankpwi  8850  rankr1g  8859  r1tskina  9806
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