uzwo3 - Metamath Proof Explorer

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Theorem uzwo3 11202

Description: Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11171 allows the lower bound

to be any real number. See also nnwo 11172 and nnwos 11174. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.)

**Assertion**
Ref	Expression
uzwo3	$/\$ $/\$

Distinct variable groups:

Proof of Theorem uzwo3 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		renegcl 9901	. . . 4
2	1	adantr 465	. . 3 $/\$ $/\$
3		arch 10813	. . 3
4	2, 3	syl 16	. 2 $/\$ $/\$
5		simplrl 761	. . . . 5 $/\$ $/\$ $/\$ $/\$
6		simplrl 761	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
7		nnnegz 10888	. . . . . . . . . . . . 13
8	6, 7	syl 16	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
9	8	zred 10990	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
10		simprl 756	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
11	10	zred 10990	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
12		simpll 753	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
13	6	nnred 10571	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
14		simplrr 762	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	12, 13, 14	ltnegcon1d 10153	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		simprr 757	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
17	9, 12, 11, 15, 16	ltletrd 9759	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	9, 11, 17	ltled 9750	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		eluz 11119	. . . . . . . . . . 11 $/\$
20	8, 10, 19	syl2anc 661	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21	18, 20	mpbird 232	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
22	21	expr 615	. . . . . . . 8 $/\$ $/\$ $/\$
23	22	ralrimiva 2871	. . . . . . 7 $/\$ $/\$
24		rabss 3573	. . . . . . 7
25	23, 24	sylibr 212	. . . . . 6 $/\$ $/\$
26	25	adantlr 714	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	5, 26	sstrd 3509	. . . 4 $/\$ $/\$ $/\$ $/\$
28		simplrr 762	. . . 4 $/\$ $/\$ $/\$ $/\$
29		infmssuzcl 11190	. . . 4 $/\$
30	27, 28, 29	syl2anc 661	. . 3 $/\$ $/\$ $/\$ $/\$
31		infmssuzle 11189	. . . . 5 $/\$
32	27, 31	sylan 471	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
33	32	ralrimiva 2871	. . 3 $/\$ $/\$ $/\$ $/\$
34	30	adantr 465	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simprr 757	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		breq2 4460	. . . . . . . 8
37	36	rspcv 3206	. . . . . . 7
38	34, 35, 37	sylc 60	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	27	adantr 465	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		simprl 756	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		infmssuzle 11189	. . . . . . 7 $/\$
42	39, 40, 41	syl2anc 661	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		uzssz 11125	. . . . . . . . . . 11
44		zssre 10892	. . . . . . . . . . 11
45	43, 44	sstri 3508	. . . . . . . . . 10
46	27, 45	syl6ss 3511	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
47	46	adantr 465	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	47, 40	sseldd 3500	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	46, 30	sseldd 3500	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50	49	adantr 465	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	48, 50	letri3d 9744	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	38, 42, 51	mpbir2and 922	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52	expr 615	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	ralrimiva 2871	. . 3 $/\$ $/\$ $/\$ $/\$
55		breq1 4459	. . . . 5
56	55	ralbidv 2896	. . . 4
57	56	eqreu 3291	. . 3 $/\$ $/\$
58	30, 33, 54, 57	syl3anc 1228	. 2 $/\$ $/\$ $/\$ $/\$
59	4, 58	rexlimddv 2953	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 184 $/\$ wa 369

wceq 1395

wcel 1819

wne 2652

wral 2807

wrex 2808

wreu 2809

crab 2811

wss 3471

c0 3793 class class class wbr 4456

ccnv 5007

cfv 5594

csup 7918

cr 9508

clt 9645

cle 9646

cneg 9825

cn 10556

cz 10885

cuz 11106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1619 ax-4 1632 ax-5 1705 ax-6 1748 ax-7 1791 ax-8 1821 ax-9 1823 ax-10 1838 ax-11 1843 ax-12 1855 ax-13 2000 ax-ext 2435 ax-sep 4578 ax-nul 4586 ax-pow 4634 ax-pr 4695 ax-un 6591 ax-cnex 9565 ax-resscn 9566 ax-1cn 9567 ax-icn 9568 ax-addcl 9569 ax-addrcl 9570 ax-mulcl 9571 ax-mulrcl 9572 ax-mulcom 9573 ax-addass 9574 ax-mulass 9575 ax-distr 9576 ax-i2m1 9577 ax-1ne0 9578 ax-1rid 9579 ax-rnegex 9580 ax-rrecex 9581 ax-cnre 9582 ax-pre-lttri 9583 ax-pre-lttrn 9584 ax-pre-ltadd 9585 ax-pre-mulgt0 9586 ax-pre-sup 9587

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1614 df-nf 1618 df-sb 1741 df-eu 2287 df-mo 2288 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-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3431 df-dif 3474 df-un 3476 df-in 3478 df-ss 3485 df-pss 3487 df-nul 3794 df-if 3945 df-pw 4017 df-sn 4033 df-pr 4035 df-tp 4037 df-op 4039 df-uni 4252 df-iun 4334 df-br 4457 df-opab 4516 df-mpt 4517 df-tr 4551 df-eprel 4800 df-id 4804 df-po 4809 df-so 4810 df-fr 4847 df-we 4849 df-ord 4890 df-on 4891 df-lim 4892 df-suc 4893 df-xp 5014 df-rel 5015 df-cnv 5016 df-co 5017 df-dm 5018 df-rn 5019 df-res 5020 df-ima 5021 df-iota 5557 df-fun 5596 df-fn 5597 df-f 5598 df-f1 5599 df-fo 5600 df-f1o 5601 df-fv 5602 df-riota 6258 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6700 df-recs 7060 df-rdg 7094 df-er 7329 df-en 7536 df-dom 7537 df-sdom 7538 df-sup 7919 df-pnf 9647 df-mnf 9648 df-xr 9649 df-ltxr 9650 df-le 9651 df-sub 9826 df-neg 9827 df-nn 10557 df-n0 10817 df-z 10886 df-uz 11107

This theorem is referenced by: zmin 11203

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