Theorem rankr1b 4699

Description: A relationship between rank and R1. See rankr1a 4677 for the membership version.

Hypothesis

Ref	Expression
rankr1b.1	|- A e. V

Assertion

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

Proof of Theorem rankr1b

Step	Hyp	Ref	Expression
1		rankr1id 4697	. . . . 5 |- (B e. On <-> (rank` (R1` B)) = B)
2	1	biimp 151	. . . 4 |- (B e. On -> (rank` (R1` B)) = B)
3	2	sseq2d 2089	. . 3 |- (B e. On -> ((rank` A) (_ (rank` (R1` B)) <-> (rank` A) (_ B))
4		fvex 3732	. . . 4 |- (R1` B) e. V
5	4	rankss 4688	. . 3 |- (A (_ (R1` B) -> (rank` A) (_ (rank` (R1` B)))
6	3, 5	syl5bi 208	. 2 |- (B e. On -> (A (_ (R1` B) -> (rank` A) (_ B))
7		rankon 4671	. . . 4 |- (rank` A) e. On
8		r1ord3 4657	. . . 4 |- (((rank` A) e. On /\ B e. On) -> ((rank` A) (_ B -> (R1` (rank` A)) (_ (R1` B)))
9	7, 8	mpan 695	. . 3 |- (B e. On -> ((rank` A) (_ B -> (R1` (rank` A)) (_ (R1` B)))
10		rankr1b.1	. . . . 5 |- A e. V
11		r1rankid 4694	. . . . 5 |- (A e. V -> A (_ (R1` (rank` A)))
12	10, 11	ax-mp 7	. . . 4 |- A (_ (R1` (rank` A))
13		sstr 2072	. . . 4 |- ((A (_ (R1` (rank` A)) /\ (R1` (rank` A)) (_ (R1` B)) -> A (_ (R1` B))
14	12, 13	mpan 695	. . 3 |- ((R1` (rank` A)) (_ (R1` B) -> A (_ (R1` B))
15	9, 14	syl6 22	. 2 |- (B e. On -> ((rank` A) (_ B -> A (_ (R1` B)))
16	6, 15	impbid 516	1 |- (B e. On -> (A (_ (R1` B) <-> (rank` A) (_ B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  Oncon0 2948  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

Copyright terms: Public domain