Theorem smoiso 7052

Description: If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)

Assertion

Ref	Expression
smoiso

Proof of Theorem smoiso

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isof1o 6221	. . . 4
2		f1of 5821	. . . 4
3	1, 2	syl 16	. . 3
4		ffdm 5750	. . . . . 6
5	4	simpld 459	. . . . 5
6		fss 5744	. . . . 5
7	5, 6	sylan 471	. . . 4
8	7	3adant2 1015	. . 3
9	3, 8	syl3an1 1261	. 2
10		fdm 5740	. . . . . 6
11	10	eqcomd 2465	. . . . 5
12		ordeq 4890	. . . . 5
13	1, 2, 11, 12	4syl 21	. . . 4
14	13	biimpa 484	. . 3
15	14	3adant3 1016	. 2
16	10	eleq2d 2527	. . . . . . 7
17	10	eleq2d 2527	. . . . . . 7
18	16, 17	anbi12d 710	. . . . . 6
19	1, 2, 18	3syl 20	. . . . 5
20		isorel 6222	. . . . . . . 8
21		epel 4799	. . . . . . . 8
22		fvex 5881	. . . . . . . . 9
23	22	epelc 4798	. . . . . . . 8
24	20, 21, 23	3bitr3g 287	. . . . . . 7
25	24	biimpd 207	. . . . . 6
26	25	ex 434	. . . . 5
27	19, 26	sylbid 215	. . . 4
28	27	ralrimivv 2877	. . 3
29	28	3ad2ant1 1017	. 2
30		df-smo 7036	. 2
31	9, 15, 29, 30	syl3anbrc 1180	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  C_wss 3475   class class class wbr 4452  cep 4794  Ordword 4882  con0 4883  domcdm 5004  -->wf 5589  -1-1-onto->wf1o 5592  `cfv 5593  Isomwiso 5594  Smowsmo 7035

This theorem is referenced by:  smoiso2  7059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-iota 5556  df-fn 5596  df-f 5597  df-f1 5598  df-f1o 5600  df-fv 5601  df-isom 5602  df-smo 7036