Theorem rankr1b 6111

Description: A relationship between rank and R1. See rankr1a 6089 for the membership version.

Hypothesis

Ref	Expression
rankr1b.1	|- A e. _V

Assertion

Ref	Expression
rankr1b	|- (B e. On -> (A C_ (R1` B) <-> (rank` A) C_ B))

Proof of Theorem rankr1b

Step	Hyp	Ref	Expression
1		fvex 4817	. . . 4 |- (R1` B) e. _V
2	1	rankss 6100	. . 3 |- (A C_ (R1` B) -> (rank` A) C_ (rank` (R1` B)))
3		rankr1id 6109	. . . . 5 |- (B e. On <-> (rank` (R1` B)) = B)
4	3	biimpi 236	. . . 4 |- (B e. On -> (rank` (R1` B)) = B)
5	4	sseq2d 2904	. . 3 |- (B e. On -> ((rank` A) C_ (rank` (R1` B)) <-> (rank` A) C_ B))
6	2, 5	syl5ib 274	. 2 |- (B e. On -> (A C_ (R1` B) -> (rank` A) C_ B))
7		rankon 6069	. . . 4 |- (rank` A) e. On
8		r1ord3 6054	. . . 4 |- (((rank` A) e. On /\ B e. On) -> ((rank` A) C_ B -> (R1` (rank` A)) C_ (R1` B)))
9	7, 8	mpan 773	. . 3 |- (B e. On -> ((rank` A) C_ B -> (R1` (rank` A)) C_ (R1` B)))
10		rankr1b.1	. . . . 5 |- A e. _V
11		r1rankid 6106	. . . . 5 |- (A e. _V -> A C_ (R1` (rank` A)))
12	10, 11	ax-mp 7	. . . 4 |- A C_ (R1` (rank` A))
13		sstr 2887	. . . 4 |- ((A C_ (R1` (rank` A)) /\ (R1` (rank` A)) C_ (R1` B)) -> A C_ (R1` B))
14	12, 13	mpan 773	. . 3 |- ((R1` (rank` A)) C_ (R1` B) -> A C_ (R1` B))
15	9, 14	syl6 39	. 2 |- (B e. On -> ((rank` A) C_ B -> A C_ (R1` B)))
16	6, 15	impbid 250	1 |- (B e. On -> (A C_ (R1` B) <-> (rank` A) C_ B))

Syntax hints:   -> wi 3   <-> wb 231   = wceq 1615   e. wcel 1617  _Vcvv 2569   C_ wss 2859  Oncon0 3843  ` cfv 4163  R1cr1 6029  rankcrnk 6030

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-13 1628  ax-14 1629  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152  ax-rep 3628  ax-sep 3638  ax-nul 3645  ax-pow 3681  ax-pr 3719  ax-un 3961  ax-reg 5972  ax-inf2 6008

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-3or 1131  df-3an 1132  df-ex 1645  df-sb 1845  df-eu 2070  df-mo 2071  df-clab 2158  df-cleq 2163  df-clel 2166  df-ne 2297  df-ral 2389  df-rex 2390  df-rab 2392  df-v 2571  df-sbc 2731  df-csb 2806  df-dif 2862  df-un 2864  df-in 2866  df-ss 2868  df-pss 2870  df-nul 3115  df-if 3213  df-pw 3261  df-sn 3274  df-pr 3275  df-tp 3277  df-op 3278  df-uni 3399  df-int 3433  df-iun 3470  df-br 3540  df-opab 3598  df-tr 3612  df-eprel 3776  df-id 3779  df-po 3784  df-so 3796  df-fr 3814  df-we 3830  df-ord 3846  df-on 3847  df-lim 3848  df-suc 3849  df-om 4118  df-xp 4165  df-rel 4166  df-cnv 4167  df-co 4168  df-dm 4169  df-rn 4170  df-res 4171  df-ima 4172  df-fun 4173  df-fn 4174  df-fv 4179  df-rdg 5344  df-r1 6031  df-rank 6032

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