Theorem indcardi 8864

Description: Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)

Hypotheses

Ref	Expression
indcardi.a	|- ( ph -> A e. V )
indcardi.b	|- ( ph -> T e. dom card )
indcardi.c	|- ( ( ph /\ R ~<_ T /\ A. y ( S ~< R -> ch ) ) -> ps )
indcardi.d	|- ( x = y -> ( ps <-> ch ) )
indcardi.e	|- ( x = A -> ( ps <-> th ) )
indcardi.f	|- ( x = y -> R = S )
indcardi.g	|- ( x = A -> R = T )

Assertion

Ref	Expression
indcardi	|- ( ph -> th )

Distinct variable groups:   x, y, T   x, A   x, S   ch, x   ph, x, y   th, x   y, R   ps, y

Allowed substitution hints:   ps( x)   ch( y)   th( y)   A( y)   R( x)   S( y)   V( x, y)

Proof of Theorem indcardi

Step	Hyp	Ref	Expression
1		indcardi.b	. . 3 |- ( ph -> T e. dom card )
2		domrefg 7990	. . 3 |- ( T e. dom card -> T ~<_ T )
3	1, 2	syl 17	. 2 |- ( ph -> T ~<_ T )
4		indcardi.a	. . 3 |- ( ph -> A e. V )
5		cardon 8770	. . . 4 |- ( card ` T ) e. On
6	5	a1i 11	. . 3 |- ( ph -> ( card ` T ) e. On )
7		simpl1 1064	. . . . 5 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) ) /\ R ~<_ T ) -> ph )
8		simpr 477	. . . . 5 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) ) /\ R ~<_ T ) -> R ~<_ T )
9		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> S ~< R )
10		simpl1 1064	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> ph )
11	10, 1	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> T e. dom card )
12		sdomdom 7983	. . . . . . . . . . . . . . . . 17 |- ( S ~< R -> S ~<_ R )
13	12	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> S ~<_ R )
14		simpl3 1066	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> R ~<_ T )
15		domtr 8009	. . . . . . . . . . . . . . . 16 |- ( ( S ~<_ R /\ R ~<_ T ) -> S ~<_ T )
16	13, 14, 15	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> S ~<_ T )
17		numdom 8861	. . . . . . . . . . . . . . 15 |- ( ( T e. dom card /\ S ~<_ T ) -> S e. dom card )
18	11, 16, 17	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> S e. dom card )
19		numdom 8861	. . . . . . . . . . . . . . 15 |- ( ( T e. dom card /\ R ~<_ T ) -> R e. dom card )
20	11, 14, 19	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> R e. dom card )
21		cardsdom2 8814	. . . . . . . . . . . . . 14 |- ( ( S e. dom card /\ R e. dom card ) -> ( ( card ` S ) e. ( card ` R ) <-> S ~< R ) )
22	18, 20, 21	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> ( ( card ` S ) e. ( card ` R ) <-> S ~< R ) )
23	9, 22	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> ( card ` S ) e. ( card ` R ) )
24		id 22	. . . . . . . . . . . . 13 |- ( ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) )
25	24	com3l 89	. . . . . . . . . . . 12 |- ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ( ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> ch ) ) )
26	23, 16, 25	sylc 65	. . . . . . . . . . 11 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) /\ S ~< R ) -> ( ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> ch ) )
27	26	ex 450	. . . . . . . . . 10 |- ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) -> ( S ~< R -> ( ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> ch ) ) )
28	27	com23 86	. . . . . . . . 9 |- ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) -> ( ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> ( S ~< R -> ch ) ) )
29	28	alimdv 1845	. . . . . . . 8 |- ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ R ~<_ T ) -> ( A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> A. y ( S ~< R -> ch ) ) )
30	29	3exp 1264	. . . . . . 7 |- ( ph -> ( ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) -> ( R ~<_ T -> ( A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> A. y ( S ~< R -> ch ) ) ) ) )
31	30	com34 91	. . . . . 6 |- ( ph -> ( ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) -> ( A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) -> ( R ~<_ T -> A. y ( S ~< R -> ch ) ) ) ) )
32	31	3imp1 1280	. . . . 5 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) ) /\ R ~<_ T ) -> A. y ( S ~< R -> ch ) )
33		indcardi.c	. . . . 5 |- ( ( ph /\ R ~<_ T /\ A. y ( S ~< R -> ch ) ) -> ps )
34	7, 8, 32, 33	syl3anc 1326	. . . 4 |- ( ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) ) /\ R ~<_ T ) -> ps )
35	34	ex 450	. . 3 |- ( ( ph /\ ( ( card ` R ) e. On /\ ( card ` R ) C_ ( card ` T ) ) /\ A. y ( ( card ` S ) e. ( card ` R ) -> ( S ~<_ T -> ch ) ) ) -> ( R ~<_ T -> ps ) )
36		indcardi.f	. . . . 5 |- ( x = y -> R = S )
37	36	breq1d 4663	. . . 4 |- ( x = y -> ( R ~<_ T <-> S ~<_ T ) )
38		indcardi.d	. . . 4 |- ( x = y -> ( ps <-> ch ) )
39	37, 38	imbi12d 334	. . 3 |- ( x = y -> ( ( R ~<_ T -> ps ) <-> ( S ~<_ T -> ch ) ) )
40		indcardi.g	. . . . 5 |- ( x = A -> R = T )
41	40	breq1d 4663	. . . 4 |- ( x = A -> ( R ~<_ T <-> T ~<_ T ) )
42		indcardi.e	. . . 4 |- ( x = A -> ( ps <-> th ) )
43	41, 42	imbi12d 334	. . 3 |- ( x = A -> ( ( R ~<_ T -> ps ) <-> ( T ~<_ T -> th ) ) )
44	36	fveq2d 6195	. . 3 |- ( x = y -> ( card ` R ) = ( card ` S ) )
45	40	fveq2d 6195	. . 3 |- ( x = A -> ( card ` R ) = ( card ` T ) )
46	4, 6, 35, 39, 43, 44, 45	tfisi 7058	. 2 |- ( ph -> ( T ~<_ T -> th ) )
47	3, 46	mpd 15	1 |- ( ph -> th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   ~<_ 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

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-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-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  uzindi  12781  symggen  17890