Theorem rankr1b 5810

Description: A relationship between rank and R1. See rankr1a 5788 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		rankr1id 5808	. . . . 5 |- (B e. On <-> (rank` (R1` B)) = B)
2	1	biimpi 168	. . . 4 |- (B e. On -> (rank` (R1` B)) = B)
3	2	sseq2d 2645	. . 3 |- (B e. On -> ((rank` A) C_ (rank` (R1` B)) <-> (rank` A) C_ B))
4		fvex 4689	. . . 4 |- (R1` B) e. _V
5	4	rankss 5799	. . 3 |- (A C_ (R1` B) -> (rank` A) C_ (rank` (R1` B)))
6	3, 5	syl5bi 225	. 2 |- (B e. On -> (A C_ (R1` B) -> (rank` A) C_ B))
7		rankon 5782	. . . 4 |- (rank` A) e. On
8		r1ord3 5768	. . . 4 |- (((rank` A) e. On /\ B e. On) -> ((rank` A) C_ B -> (R1` (rank` A)) C_ (R1` B)))
9	7, 8	mpan 759	. . 3 |- (B e. On -> ((rank` A) C_ B -> (R1` (rank` A)) C_ (R1` B)))
10		rankr1b.1	. . . . 5 |- A e. _V
11		r1rankid 5805	. . . . 5 |- (A e. _V -> A C_ (R1` (rank` A)))
12	10, 11	ax-mp 7	. . . 4 |- A C_ (R1` (rank` A))
13		sstr 2625	. . . 4 |- ((A C_ (R1` (rank` A)) /\ (R1` (rank` A)) C_ (R1` B)) -> A C_ (R1` B))
14	12, 13	mpan 759	. . 3 |- ((R1` (rank` A)) C_ (R1` B) -> A C_ (R1` B))
15	9, 14	syl6 25	. 2 |- (B e. On -> ((rank` A) C_ B -> A C_ (R1` B)))
16	6, 15	impbid 574	1 |- (B e. On -> (A C_ (R1` B) <-> (rank` A) C_ B))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  Oncon0 3657  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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