Theorem rankopb 8715

Description: The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
rankopb	|- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` <. A , B >. ) = suc suc ( ( rank ` A ) u. ( rank ` B ) ) )

Proof of Theorem rankopb

Step	Hyp	Ref	Expression
1		dfopg 4400	. . 3 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> <. A , B >. = { { A } , { A , B } } )
2	1	fveq2d 6195	. 2 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` <. A , B >. ) = ( rank ` { { A } , { A , B } } ) )
3		snwf 8672	. . . 4 |- ( A e. U. ( R1 " On ) -> { A } e. U. ( R1 " On ) )
4	3	adantr 481	. . 3 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> { A } e. U. ( R1 " On ) )
5		prwf 8674	. . 3 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> { A , B } e. U. ( R1 " On ) )
6		rankprb 8714	. . 3 |- ( ( { A } e. U. ( R1 " On ) /\ { A , B } e. U. ( R1 " On ) ) -> ( rank ` { { A } , { A , B } } ) = suc ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) )
7	4, 5, 6	syl2anc 693	. 2 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` { { A } , { A , B } } ) = suc ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) )
8		snsspr1 4345	. . . . . 6 |- { A } C_ { A , B }
9		ssequn1 3783	. . . . . 6 |- ( { A } C_ { A , B } <-> ( { A } u. { A , B } ) = { A , B } )
10	8, 9	mpbi 220	. . . . 5 |- ( { A } u. { A , B } ) = { A , B }
11	10	fveq2i 6194	. . . 4 |- ( rank ` ( { A } u. { A , B } ) ) = ( rank ` { A , B } )
12		rankunb 8713	. . . . 5 |- ( ( { A } e. U. ( R1 " On ) /\ { A , B } e. U. ( R1 " On ) ) -> ( rank ` ( { A } u. { A , B } ) ) = ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) )
13	4, 5, 12	syl2anc 693	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` ( { A } u. { A , B } ) ) = ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) )
14		rankprb 8714	. . . 4 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` { A , B } ) = suc ( ( rank ` A ) u. ( rank ` B ) ) )
15	11, 13, 14	3eqtr3a 2680	. . 3 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) = suc ( ( rank ` A ) u. ( rank ` B ) ) )
16		suceq 5790	. . 3 |- ( ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) = suc ( ( rank ` A ) u. ( rank ` B ) ) -> suc ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) = suc suc ( ( rank ` A ) u. ( rank ` B ) ) )
17	15, 16	syl 17	. 2 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> suc ( ( rank ` { A } ) u. ( rank ` { A , B } ) ) = suc suc ( ( rank ` A ) u. ( rank ` B ) ) )
18	2, 7, 17	3eqtrd 2660	1 |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` <. A , B >. ) = suc suc ( ( rank ` A ) u. ( rank ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   U.cuni 4436   "cima 5117   Oncon0 5723   suc csuc 5725   ` 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:  rankop  8721