Theorem rankmapu 8741

Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)

Hypotheses

Ref	Expression
rankxpl.1	|- A e. _V
rankxpl.2	|- B e. _V

Assertion

Ref	Expression
rankmapu	|- ( rank ` ( A ^m B ) ) C_ suc suc suc ( rank ` ( A u. B ) )

Proof of Theorem rankmapu

Step	Hyp	Ref	Expression
1		mapsspw 7893	. . 3 |- ( A ^m B ) C_ ~P ( B X. A )
2		rankxpl.2	. . . . . 6 |- B e. _V
3		rankxpl.1	. . . . . 6 |- A e. _V
4	2, 3	xpex 6962	. . . . 5 |- ( B X. A ) e. _V
5	4	pwex 4848	. . . 4 |- ~P ( B X. A ) e. _V
6	5	rankss 8712	. . 3 |- ( ( A ^m B ) C_ ~P ( B X. A ) -> ( rank ` ( A ^m B ) ) C_ ( rank ` ~P ( B X. A ) ) )
7	1, 6	ax-mp 5	. 2 |- ( rank ` ( A ^m B ) ) C_ ( rank ` ~P ( B X. A ) )
8	4	rankpw 8706	. . 3 |- ( rank ` ~P ( B X. A ) ) = suc ( rank ` ( B X. A ) )
9	2, 3	rankxpu 8739	. . . . 5 |- ( rank ` ( B X. A ) ) C_ suc suc ( rank ` ( B u. A ) )
10		uncom 3757	. . . . . . . 8 |- ( B u. A ) = ( A u. B )
11	10	fveq2i 6194	. . . . . . 7 |- ( rank ` ( B u. A ) ) = ( rank ` ( A u. B ) )
12		suceq 5790	. . . . . . 7 |- ( ( rank ` ( B u. A ) ) = ( rank ` ( A u. B ) ) -> suc ( rank ` ( B u. A ) ) = suc ( rank ` ( A u. B ) ) )
13	11, 12	ax-mp 5	. . . . . 6 |- suc ( rank ` ( B u. A ) ) = suc ( rank ` ( A u. B ) )
14		suceq 5790	. . . . . 6 |- ( suc ( rank ` ( B u. A ) ) = suc ( rank ` ( A u. B ) ) -> suc suc ( rank ` ( B u. A ) ) = suc suc ( rank ` ( A u. B ) ) )
15	13, 14	ax-mp 5	. . . . 5 |- suc suc ( rank ` ( B u. A ) ) = suc suc ( rank ` ( A u. B ) )
16	9, 15	sseqtri 3637	. . . 4 |- ( rank ` ( B X. A ) ) C_ suc suc ( rank ` ( A u. B ) )
17		rankon 8658	. . . . . 6 |- ( rank ` ( B X. A ) ) e. On
18	17	onordi 5832	. . . . 5 |- Ord ( rank ` ( B X. A ) )
19		rankon 8658	. . . . . . . 8 |- ( rank ` ( A u. B ) ) e. On
20	19	onsuci 7038	. . . . . . 7 |- suc ( rank ` ( A u. B ) ) e. On
21	20	onsuci 7038	. . . . . 6 |- suc suc ( rank ` ( A u. B ) ) e. On
22	21	onordi 5832	. . . . 5 |- Ord suc suc ( rank ` ( A u. B ) )
23		ordsucsssuc 7023	. . . . 5 |- ( ( Ord ( rank ` ( B X. A ) ) /\ Ord suc suc ( rank ` ( A u. B ) ) ) -> ( ( rank ` ( B X. A ) ) C_ suc suc ( rank ` ( A u. B ) ) <-> suc ( rank ` ( B X. A ) ) C_ suc suc suc ( rank ` ( A u. B ) ) ) )
24	18, 22, 23	mp2an 708	. . . 4 |- ( ( rank ` ( B X. A ) ) C_ suc suc ( rank ` ( A u. B ) ) <-> suc ( rank ` ( B X. A ) ) C_ suc suc suc ( rank ` ( A u. B ) ) )
25	16, 24	mpbi 220	. . 3 |- suc ( rank ` ( B X. A ) ) C_ suc suc suc ( rank ` ( A u. B ) )
26	8, 25	eqsstri 3635	. 2 |- ( rank ` ~P ( B X. A ) ) C_ suc suc suc ( rank ` ( A u. B ) )
27	7, 26	sstri 3612	1 |- ( rank ` ( A ^m B ) ) C_ suc suc suc ( rank ` ( A u. B ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   Ord word 5722   suc csuc 5725   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-pm 7860  df-r1 8627  df-rank 8628

This theorem is referenced by: (None)