Theorem rankelop 8737

Description: Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)

Hypotheses

Ref	Expression
rankelun.1	|- A e. _V
rankelun.2	|- B e. _V
rankelun.3	|- C e. _V
rankelun.4	|- D e. _V

Assertion

Ref	Expression
rankelop	|- ( ( ( rank ` A ) e. ( rank ` C ) /\ ( rank ` B ) e. ( rank ` D ) ) -> ( rank ` <. A , B >. ) e. ( rank ` <. C , D >. ) )

Proof of Theorem rankelop

Step	Hyp	Ref	Expression
1		rankelun.1	. . . 4 |- A e. _V
2		rankelun.2	. . . 4 |- B e. _V
3		rankelun.3	. . . 4 |- C e. _V
4		rankelun.4	. . . 4 |- D e. _V
5	1, 2, 3, 4	rankelpr 8736	. . 3 |- ( ( ( rank ` A ) e. ( rank ` C ) /\ ( rank ` B ) e. ( rank ` D ) ) -> ( rank ` { A , B } ) e. ( rank ` { C , D } ) )
6		rankon 8658	. . . . 5 |- ( rank ` { C , D } ) e. On
7	6	onordi 5832	. . . 4 |- Ord ( rank ` { C , D } )
8		ordsucelsuc 7022	. . . 4 |- ( Ord ( rank ` { C , D } ) -> ( ( rank ` { A , B } ) e. ( rank ` { C , D } ) <-> suc ( rank ` { A , B } ) e. suc ( rank ` { C , D } ) ) )
9	7, 8	ax-mp 5	. . 3 |- ( ( rank ` { A , B } ) e. ( rank ` { C , D } ) <-> suc ( rank ` { A , B } ) e. suc ( rank ` { C , D } ) )
10	5, 9	sylib 208	. 2 |- ( ( ( rank ` A ) e. ( rank ` C ) /\ ( rank ` B ) e. ( rank ` D ) ) -> suc ( rank ` { A , B } ) e. suc ( rank ` { C , D } ) )
11	1, 2	rankop 8721	. . 3 |- ( rank ` <. A , B >. ) = suc suc ( ( rank ` A ) u. ( rank ` B ) )
12	1, 2	rankpr 8720	. . . 4 |- ( rank ` { A , B } ) = suc ( ( rank ` A ) u. ( rank ` B ) )
13		suceq 5790	. . . 4 |- ( ( rank ` { A , B } ) = suc ( ( rank ` A ) u. ( rank ` B ) ) -> suc ( rank ` { A , B } ) = suc suc ( ( rank ` A ) u. ( rank ` B ) ) )
14	12, 13	ax-mp 5	. . 3 |- suc ( rank ` { A , B } ) = suc suc ( ( rank ` A ) u. ( rank ` B ) )
15	11, 14	eqtr4i 2647	. 2 |- ( rank ` <. A , B >. ) = suc ( rank ` { A , B } )
16	3, 4	rankop 8721	. . 3 |- ( rank ` <. C , D >. ) = suc suc ( ( rank ` C ) u. ( rank ` D ) )
17	3, 4	rankpr 8720	. . . 4 |- ( rank ` { C , D } ) = suc ( ( rank ` C ) u. ( rank ` D ) )
18		suceq 5790	. . . 4 |- ( ( rank ` { C , D } ) = suc ( ( rank ` C ) u. ( rank ` D ) ) -> suc ( rank ` { C , D } ) = suc suc ( ( rank ` C ) u. ( rank ` D ) ) )
19	17, 18	ax-mp 5	. . 3 |- suc ( rank ` { C , D } ) = suc suc ( ( rank ` C ) u. ( rank ` D ) )
20	16, 19	eqtr4i 2647	. 2 |- ( rank ` <. C , D >. ) = suc ( rank ` { C , D } )
21	10, 15, 20	3eltr4g 2718	1 |- ( ( ( rank ` A ) e. ( rank ` C ) /\ ( rank ` B ) e. ( rank ` D ) ) -> ( rank ` <. A , B >. ) e. ( rank ` <. C , D >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {cpr 4179   <.cop 4183   Ord word 5722   suc csuc 5725   ` cfv 5888   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

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: (None)