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Theorem rankr1a 8870
Description: A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 8869 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8898 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 8869 . . 3 (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1𝐵)))
3 rankon 8829 . . . 4 (rank‘𝐴) ∈ On
4 ontri1 5916 . . . 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 2137  Vcvv 3338  wss 3713  Oncon0 5882  cfv 6047  𝑅1cr1 8796  rankcrnk 8797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-reg 8660  ax-inf2 8709
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-om 7229  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-r1 8798  df-rank 8799
This theorem is referenced by:  r1val2  8871  r1pwALT  8880  elhf2  32586
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