Theorem infunsdom1 9035

Description: The union of two sets that are strictly dominated by the infinite set X is also dominated by X. This version of infunsdom 9036 assumes additionally that A is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
infunsdom1	|- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> ( A u. B ) ~< X )

Proof of Theorem infunsdom1

Step	Hyp	Ref	Expression
1		simprl 794	. . . . 5 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> A ~<_ B )
2		domsdomtr 8095	. . . . 5 |- ( ( A ~<_ B /\ B ~< om ) -> A ~< om )
3	1, 2	sylan 488	. . . 4 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ B ~< om ) -> A ~< om )
4		unfi2 8229	. . . 4 |- ( ( A ~< om /\ B ~< om ) -> ( A u. B ) ~< om )
5	3, 4	sylancom 701	. . 3 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ B ~< om ) -> ( A u. B ) ~< om )
6		simpllr 799	. . 3 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ B ~< om ) -> om ~<_ X )
7		sdomdomtr 8093	. . 3 |- ( ( ( A u. B ) ~< om /\ om ~<_ X ) -> ( A u. B ) ~< X )
8	5, 6, 7	syl2anc 693	. 2 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ B ~< om ) -> ( A u. B ) ~< X )
9		omelon 8543	. . . . . 6 |- om e. On
10		onenon 8775	. . . . . 6 |- ( om e. On -> om e. dom card )
11	9, 10	ax-mp 5	. . . . 5 |- om e. dom card
12		simpll 790	. . . . . 6 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> X e. dom card )
13		sdomdom 7983	. . . . . . 7 |- ( B ~< X -> B ~<_ X )
14	13	ad2antll 765	. . . . . 6 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> B ~<_ X )
15		numdom 8861	. . . . . 6 |- ( ( X e. dom card /\ B ~<_ X ) -> B e. dom card )
16	12, 14, 15	syl2anc 693	. . . . 5 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> B e. dom card )
17		domtri2 8815	. . . . 5 |- ( ( om e. dom card /\ B e. dom card ) -> ( om ~<_ B <-> -. B ~< om ) )
18	11, 16, 17	sylancr 695	. . . 4 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> ( om ~<_ B <-> -. B ~< om ) )
19	18	biimpar 502	. . 3 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ -. B ~< om ) -> om ~<_ B )
20		uncom 3757	. . . . 5 |- ( A u. B ) = ( B u. A )
21	16	adantr 481	. . . . . 6 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> B e. dom card )
22		simpr 477	. . . . . 6 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> om ~<_ B )
23	1	adantr 481	. . . . . 6 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> A ~<_ B )
24		infunabs 9029	. . . . . 6 |- ( ( B e. dom card /\ om ~<_ B /\ A ~<_ B ) -> ( B u. A ) ~~ B )
25	21, 22, 23, 24	syl3anc 1326	. . . . 5 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> ( B u. A ) ~~ B )
26	20, 25	syl5eqbr 4688	. . . 4 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> ( A u. B ) ~~ B )
27		simplrr 801	. . . 4 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> B ~< X )
28		ensdomtr 8096	. . . 4 |- ( ( ( A u. B ) ~~ B /\ B ~< X ) -> ( A u. B ) ~< X )
29	26, 27, 28	syl2anc 693	. . 3 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ om ~<_ B ) -> ( A u. B ) ~< X )
30	19, 29	syldan 487	. 2 |- ( ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) /\ -. B ~< om ) -> ( A u. B ) ~< X )
31	8, 30	pm2.61dan 832	1 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( A ~<_ B /\ B ~< X ) ) -> ( A u. B ) ~< X )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   u. cun 3572   class class class wbr 4653   dom cdm 5114   Oncon0 5723   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761

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-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-rmo 2920  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-se 5074  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-isom 5897  df-riota 6611  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-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  infunsdom  9036