Theorem sucxpdom 8169

Description: Cartesian product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
sucxpdom	|- ( 1o ~< A -> suc A ~<_ ( A X. A ) )

Proof of Theorem sucxpdom

Step	Hyp	Ref	Expression
1		df-suc 5729	. 2 |- suc A = ( A u. { A } )
2		relsdom 7962	. . . . . . . . 9 |- Rel ~<
3	2	brrelex2i 5159	. . . . . . . 8 |- ( 1o ~< A -> A e. _V )
4		1on 7567	. . . . . . . 8 |- 1o e. On
5		xpsneng 8045	. . . . . . . 8 |- ( ( A e. _V /\ 1o e. On ) -> ( A X. { 1o } ) ~~ A )
6	3, 4, 5	sylancl 694	. . . . . . 7 |- ( 1o ~< A -> ( A X. { 1o } ) ~~ A )
7	6	ensymd 8007	. . . . . 6 |- ( 1o ~< A -> A ~~ ( A X. { 1o } ) )
8		endom 7982	. . . . . 6 |- ( A ~~ ( A X. { 1o } ) -> A ~<_ ( A X. { 1o } ) )
9	7, 8	syl 17	. . . . 5 |- ( 1o ~< A -> A ~<_ ( A X. { 1o } ) )
10		ensn1g 8021	. . . . . . . . 9 |- ( A e. _V -> { A } ~~ 1o )
11	3, 10	syl 17	. . . . . . . 8 |- ( 1o ~< A -> { A } ~~ 1o )
12		ensdomtr 8096	. . . . . . . 8 |- ( ( { A } ~~ 1o /\ 1o ~< A ) -> { A } ~< A )
13	11, 12	mpancom 703	. . . . . . 7 |- ( 1o ~< A -> { A } ~< A )
14		0ex 4790	. . . . . . . . 9 |- (/) e. _V
15		xpsneng 8045	. . . . . . . . 9 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
16	3, 14, 15	sylancl 694	. . . . . . . 8 |- ( 1o ~< A -> ( A X. { (/) } ) ~~ A )
17	16	ensymd 8007	. . . . . . 7 |- ( 1o ~< A -> A ~~ ( A X. { (/) } ) )
18		sdomentr 8094	. . . . . . 7 |- ( ( { A } ~< A /\ A ~~ ( A X. { (/) } ) ) -> { A } ~< ( A X. { (/) } ) )
19	13, 17, 18	syl2anc 693	. . . . . 6 |- ( 1o ~< A -> { A } ~< ( A X. { (/) } ) )
20		sdomdom 7983	. . . . . 6 |- ( { A } ~< ( A X. { (/) } ) -> { A } ~<_ ( A X. { (/) } ) )
21	19, 20	syl 17	. . . . 5 |- ( 1o ~< A -> { A } ~<_ ( A X. { (/) } ) )
22		1n0 7575	. . . . . 6 |- 1o =/= (/)
23		xpsndisj 5557	. . . . . 6 |- ( 1o =/= (/) -> ( ( A X. { 1o } ) i^i ( A X. { (/) } ) ) = (/) )
24	22, 23	mp1i 13	. . . . 5 |- ( 1o ~< A -> ( ( A X. { 1o } ) i^i ( A X. { (/) } ) ) = (/) )
25		undom 8048	. . . . 5 |- ( ( ( A ~<_ ( A X. { 1o } ) /\ { A } ~<_ ( A X. { (/) } ) ) /\ ( ( A X. { 1o } ) i^i ( A X. { (/) } ) ) = (/) ) -> ( A u. { A } ) ~<_ ( ( A X. { 1o } ) u. ( A X. { (/) } ) ) )
26	9, 21, 24, 25	syl21anc 1325	. . . 4 |- ( 1o ~< A -> ( A u. { A } ) ~<_ ( ( A X. { 1o } ) u. ( A X. { (/) } ) ) )
27		sdomentr 8094	. . . . . 6 |- ( ( 1o ~< A /\ A ~~ ( A X. { 1o } ) ) -> 1o ~< ( A X. { 1o } ) )
28	7, 27	mpdan 702	. . . . 5 |- ( 1o ~< A -> 1o ~< ( A X. { 1o } ) )
29		sdomentr 8094	. . . . . 6 |- ( ( 1o ~< A /\ A ~~ ( A X. { (/) } ) ) -> 1o ~< ( A X. { (/) } ) )
30	17, 29	mpdan 702	. . . . 5 |- ( 1o ~< A -> 1o ~< ( A X. { (/) } ) )
31		unxpdom 8167	. . . . 5 |- ( ( 1o ~< ( A X. { 1o } ) /\ 1o ~< ( A X. { (/) } ) ) -> ( ( A X. { 1o } ) u. ( A X. { (/) } ) ) ~<_ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) )
32	28, 30, 31	syl2anc 693	. . . 4 |- ( 1o ~< A -> ( ( A X. { 1o } ) u. ( A X. { (/) } ) ) ~<_ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) )
33		domtr 8009	. . . 4 |- ( ( ( A u. { A } ) ~<_ ( ( A X. { 1o } ) u. ( A X. { (/) } ) ) /\ ( ( A X. { 1o } ) u. ( A X. { (/) } ) ) ~<_ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) ) -> ( A u. { A } ) ~<_ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) )
34	26, 32, 33	syl2anc 693	. . 3 |- ( 1o ~< A -> ( A u. { A } ) ~<_ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) )
35		xpen 8123	. . . 4 |- ( ( ( A X. { 1o } ) ~~ A /\ ( A X. { (/) } ) ~~ A ) -> ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) ~~ ( A X. A ) )
36	6, 16, 35	syl2anc 693	. . 3 |- ( 1o ~< A -> ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) ~~ ( A X. A ) )
37		domentr 8015	. . 3 |- ( ( ( A u. { A } ) ~<_ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) /\ ( ( A X. { 1o } ) X. ( A X. { (/) } ) ) ~~ ( A X. A ) ) -> ( A u. { A } ) ~<_ ( A X. A ) )
38	34, 36, 37	syl2anc 693	. 2 |- ( 1o ~< A -> ( A u. { A } ) ~<_ ( A X. A ) )
39	1, 38	syl5eqbr 4688	1 |- ( 1o ~< A -> suc A ~<_ ( A X. A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   Oncon0 5723   suc csuc 5725   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954

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-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-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-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-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by: (None)