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Theorem ssrankr1 8871
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 8849 . . . 4 (𝑅1 “ On) = V
31, 2eleqtrri 2838 . . 3 𝐴 (𝑅1 “ On)
4 r1fnon 8803 . . . . . 6 𝑅1 Fn On
5 fndm 6151 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
64, 5ax-mp 5 . . . . 5 dom 𝑅1 = On
76eleq2i 2831 . . . 4 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
87biimpri 218 . . 3 (𝐵 ∈ On → 𝐵 ∈ dom 𝑅1)
9 rankr1clem 8856 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
103, 8, 9sylancr 698 . 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 1632  wcel 2139  Vcvv 3340  wss 3715   cuni 4588  dom cdm 5266  cima 5269  Oncon0 5884   Fn wfn 6044  cfv 6049  𝑅1cr1 8798  rankcrnk 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-rank 8801
This theorem is referenced by:  rankr1a  8872
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