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Theorem rankr1a 8864
Description: A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 8863 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8892 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
Hypothesis
Ref Expression
rankid.1 𝐴 ∈ V
Assertion
Ref Expression
rankr1a (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))

Proof of Theorem rankr1a
StepHypRef Expression
1 rankid.1 . . . 4 𝐴 ∈ V
21ssrankr1 8863 . . 3 (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1𝐵)))
3 rankon 8823 . . . 4 (rank‘𝐴) ∈ On
4 ontri1 5910 . . . 4 ((𝐵 ∈ On ∧ (rank‘𝐴) ∈ On) → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵))
53, 4mpan2 709 . . 3 (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵))
62, 5bitr3d 270 . 2 (𝐵 ∈ On → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ ¬ (rank‘𝐴) ∈ 𝐵))
76con4bid 306 1 (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 2131  Vcvv 3332  wss 3707  Oncon0 5876  cfv 6041  𝑅1cr1 8790  rankcrnk 8791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-reg 8654  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-r1 8792  df-rank 8793
This theorem is referenced by:  r1val2  8865  r1pwALT  8874  elhf2  32580
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