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Theorem ssrankr1 8695
 Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets 𝑅1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypothesis
Ref Expression
rankid.1 𝐴 ∈ V
Assertion
Ref Expression
ssrankr1 (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1𝐵)))

Proof of Theorem ssrankr1
StepHypRef Expression
1 rankid.1 . . . 4 𝐴 ∈ V
2 unir1 8673 . . . 4 (𝑅1 “ On) = V
31, 2eleqtrri 2699 . . 3 𝐴 (𝑅1 “ On)
4 r1fnon 8627 . . . . . 6 𝑅1 Fn On
5 fndm 5988 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
64, 5ax-mp 5 . . . . 5 dom 𝑅1 = On
76eleq2i 2692 . . . 4 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
87biimpri 218 . . 3 (𝐵 ∈ On → 𝐵 ∈ dom 𝑅1)
9 rankr1clem 8680 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
103, 8, 9sylancr 695 . 2 (𝐵 ∈ On → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
1110bicomd 213 1 (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1482   ∈ wcel 1989  Vcvv 3198   ⊆ wss 3572  ∪ cuni 4434  dom cdm 5112   “ cima 5115  Oncon0 5721   Fn wfn 5881  ‘cfv 5886  𝑅1cr1 8622  rankcrnk 8623 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf2 8535 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-r1 8624  df-rank 8625 This theorem is referenced by:  rankr1a  8696
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