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Theorem tsrss 17223

Description: Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)

**Assertion**
Ref	Expression
tsrss

Proof of Theorem tsrss Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psss 17214	. . 3
2		inss1 3833	. . . . . 6
3		dmss 5323	. . . . . 6
4		ssralv 3666	. . . . . 6 $\/$ $\/$
5	2, 3, 4	mp2b 10	. . . . 5 $\/$ $\/$
6		ssralv 3666	. . . . . . 7 $\/$ $\/$
7	2, 3, 6	mp2b 10	. . . . . 6 $\/$ $\/$
8	7	ralimi 2952	. . . . 5 $\/$ $\/$
9	5, 8	syl 17	. . . 4 $\/$ $\/$
10		inss2 3834	. . . . . . . . . 10
11		dmss 5323	. . . . . . . . . 10
12	10, 11	ax-mp 5	. . . . . . . . 9
13		dmxpid 5345	. . . . . . . . 9
14	12, 13	sseqtri 3637	. . . . . . . 8
15	14	sseli 3599	. . . . . . 7
16	14	sseli 3599	. . . . . . 7
17		brinxp 5181	. . . . . . . 8 $/\$
18		brinxp 5181	. . . . . . . . 9 $/\$
19	18	ancoms 469	. . . . . . . 8 $/\$
20	17, 19	orbi12d 746	. . . . . . 7 $/\$ $\/$ $\/$
21	15, 16, 20	syl2an 494	. . . . . 6 $/\$ $\/$ $\/$
22	21	ralbidva 2985	. . . . 5 $\/$ $\/$
23	22	ralbiia 2979	. . . 4 $\/$ $\/$
24	9, 23	sylib 208	. . 3 $\/$ $\/$
25	1, 24	anim12i 590	. 2 $/\$ $\/$ $/\$ $\/$
26		eqid 2622	. . 3
27	26	istsr2 17218	. 2 $/\$ $\/$
28		eqid 2622	. . 3
29	28	istsr2 17218	. 2 $/\$ $\/$
30	25, 27, 29	3imtr4i 281	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wcel 1990

wral 2912

cin 3573

wss 3574 class class class wbr 4653

cxp 5112

cdm 5114

cps 17198

ctsr 17199

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ps 17200 df-tsr 17201

This theorem is referenced by: (None)

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