Theorem max1ALT 11416

Description: A number is less than or equal to the maximum of it and another. This version of max1 11415 omits the antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 11415 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)

Assertion

Ref	Expression
max1ALT

Proof of Theorem max1ALT

Step	Hyp	Ref	Expression
1		leid 9701	. . 3
2		iffalse 3950	. . . 4
3	2	breq2d 4464	. . 3
4	1, 3	syl5ibrcom 222	. 2
5		id 22	. . 3
6		iftrue 3947	. . 3
7	5, 6	breqtrrd 4478	. 2
8	4, 7	pm2.61d2 160	1

Syntax hints:  -.wn 3  ->wi 4  e.wcel 1818  ifcif 3941   class class class wbr 4452  cr 9512  cle 9650

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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592  ax-resscn 9570  ax-pre-lttri 9587

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-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539  df-pnf 9651  df-mnf 9652  df-xr 9653  df-ltxr 9654  df-le 9655