Theorem rankelb 8687

Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankelb	|- ( B e. U. ( R1 " On ) -> ( A e. B -> ( rank ` A ) e. ( rank ` B ) ) )

Proof of Theorem rankelb

Step	Hyp	Ref	Expression
1		r1elssi 8668	. . . . . 6 |- ( B e. U. ( R1 " On ) -> B C_ U. ( R1 " On ) )
2	1	sseld 3602	. . . . 5 |- ( B e. U. ( R1 " On ) -> ( A e. B -> A e. U. ( R1 " On ) ) )
3		rankidn 8685	. . . . 5 |- ( A e. U. ( R1 " On ) -> -. A e. ( R1 ` ( rank ` A ) ) )
4	2, 3	syl6 35	. . . 4 |- ( B e. U. ( R1 " On ) -> ( A e. B -> -. A e. ( R1 ` ( rank ` A ) ) ) )
5	4	imp 445	. . 3 |- ( ( B e. U. ( R1 " On ) /\ A e. B ) -> -. A e. ( R1 ` ( rank ` A ) ) )
6		rankon 8658	. . . . 5 |- ( rank ` B ) e. On
7		rankon 8658	. . . . 5 |- ( rank ` A ) e. On
8		ontri1 5757	. . . . 5 |- ( ( ( rank ` B ) e. On /\ ( rank ` A ) e. On ) -> ( ( rank ` B ) C_ ( rank ` A ) <-> -. ( rank ` A ) e. ( rank ` B ) ) )
9	6, 7, 8	mp2an 708	. . . 4 |- ( ( rank ` B ) C_ ( rank ` A ) <-> -. ( rank ` A ) e. ( rank ` B ) )
10		rankdmr1 8664	. . . . . 6 |- ( rank ` B ) e. dom R1
11		rankdmr1 8664	. . . . . 6 |- ( rank ` A ) e. dom R1
12		r1ord3g 8642	. . . . . 6 |- ( ( ( rank ` B ) e. dom R1 /\ ( rank ` A ) e. dom R1 ) -> ( ( rank ` B ) C_ ( rank ` A ) -> ( R1 ` ( rank ` B ) ) C_ ( R1 ` ( rank ` A ) ) ) )
13	10, 11, 12	mp2an 708	. . . . 5 |- ( ( rank ` B ) C_ ( rank ` A ) -> ( R1 ` ( rank ` B ) ) C_ ( R1 ` ( rank ` A ) ) )
14		r1rankidb 8667	. . . . . . 7 |- ( B e. U. ( R1 " On ) -> B C_ ( R1 ` ( rank ` B ) ) )
15	14	sselda 3603	. . . . . 6 |- ( ( B e. U. ( R1 " On ) /\ A e. B ) -> A e. ( R1 ` ( rank ` B ) ) )
16		ssel 3597	. . . . . 6 |- ( ( R1 ` ( rank ` B ) ) C_ ( R1 ` ( rank ` A ) ) -> ( A e. ( R1 ` ( rank ` B ) ) -> A e. ( R1 ` ( rank ` A ) ) ) )
17	15, 16	syl5com 31	. . . . 5 |- ( ( B e. U. ( R1 " On ) /\ A e. B ) -> ( ( R1 ` ( rank ` B ) ) C_ ( R1 ` ( rank ` A ) ) -> A e. ( R1 ` ( rank ` A ) ) ) )
18	13, 17	syl5 34	. . . 4 |- ( ( B e. U. ( R1 " On ) /\ A e. B ) -> ( ( rank ` B ) C_ ( rank ` A ) -> A e. ( R1 ` ( rank ` A ) ) ) )
19	9, 18	syl5bir 233	. . 3 |- ( ( B e. U. ( R1 " On ) /\ A e. B ) -> ( -. ( rank ` A ) e. ( rank ` B ) -> A e. ( R1 ` ( rank ` A ) ) ) )
20	5, 19	mt3d 140	. 2 |- ( ( B e. U. ( R1 " On ) /\ A e. B ) -> ( rank ` A ) e. ( rank ` B ) )
21	20	ex 450	1 |- ( B e. U. ( R1 " On ) -> ( A e. B -> ( rank ` A ) e. ( rank ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   U.cuni 4436   dom cdm 5114   "cima 5117   Oncon0 5723   ` cfv 5888   R1cr1 8625   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  wfelirr  8688  rankval3b  8689  rankel  8702  rankunb  8713  rankuni2b  8716  rankcf  9599