MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssrankr1 Structured version   Visualization version   GIF version

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
  Copyright terms: Public domain W3C validator