Theorem issmo 7038

Description: Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Hypotheses

Ref	Expression
issmo.1
issmo.2
issmo.3
issmo.4

Assertion

Ref	Expression
issmo

Distinct variable group:   ,,

Proof of Theorem issmo

Step	Hyp	Ref	Expression
1		issmo.1	. . 3
2		issmo.4	. . . 4
3	2	feq2i 5729	. . 3
4	1, 3	mpbir 209	. 2
5		issmo.2	. . 3
6		ordeq 4890	. . . 4
7	2, 6	ax-mp 5	. . 3
8	5, 7	mpbir 209	. 2
9	2	eleq2i 2535	. . . 4
10	2	eleq2i 2535	. . . 4
11		issmo.3	. . . 4
12	9, 10, 11	syl2anb 479	. . 3
13	12	rgen2a 2884	. 2
14		df-smo 7036	. 2
15	4, 8, 13, 14	mpbir3an 1178	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  Ordword 4882  con0 4883  domcdm 5004  -->wf 5589  `cfv 5593  Smowsmo 7035

This theorem is referenced by:  iordsmo  7047  smobeth  8982

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-in 3482  df-ss 3489  df-uni 4250  df-tr 4546  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-fn 5596  df-f 5597  df-smo 7036