sup2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sup2		Structured version Visualization version Unicode version

Theorem sup2 10979

Description: A nonempty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)

**Assertion**
Ref	Expression
sup2	$/\$ $/\$ $\/$ $/\$

Distinct variable group:

**Proof of Theorem sup2**
Step	Hyp	Ref	Expression
1		peano2re 10209	. . . . . . . . . . . 12
2	1	adantr 481	. . . . . . . . . . 11 $/\$ $\/$
3	2	a1i 11	. . . . . . . . . 10 $/\$ $\/$
4		ssel 3597	. . . . . . . . . . . . . . . 16
5		ltp1 10861	. . . . . . . . . . . . . . . . . . . . . . 23
6	1	ancli 574	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$
7		lttr 10114	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$
8	7	3expb 1266	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$
9	6, 8	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$
10	5, 9	sylan2i 687	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$
11	10	exp4b 632	. . . . . . . . . . . . . . . . . . . . 21
12	11	com34 91	. . . . . . . . . . . . . . . . . . . 20
13	12	pm2.43d 53	. . . . . . . . . . . . . . . . . . 19
14	13	imp 445	. . . . . . . . . . . . . . . . . 18 $/\$
15		breq1 4656	. . . . . . . . . . . . . . . . . . . 20
16	5, 15	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . 19
17	16	adantl 482	. . . . . . . . . . . . . . . . . 18 $/\$
18	14, 17	jaod 395	. . . . . . . . . . . . . . . . 17 $/\$ $\/$
19	18	ex 450	. . . . . . . . . . . . . . . 16 $\/$
20	4, 19	syl6 35	. . . . . . . . . . . . . . 15 $\/$
21	20	com23 86	. . . . . . . . . . . . . 14 $\/$
22	21	imp 445	. . . . . . . . . . . . 13 $/\$ $\/$
23	22	a2d 29	. . . . . . . . . . . 12 $/\$ $\/$
24	23	ralimdv2 2961	. . . . . . . . . . 11 $/\$ $\/$
25	24	expimpd 629	. . . . . . . . . 10 $/\$ $\/$
26	3, 25	jcad 555	. . . . . . . . 9 $/\$ $\/$ $/\$
27		ovex 6678	. . . . . . . . . 10
28		eleq1 2689	. . . . . . . . . . 11
29		breq2 4657	. . . . . . . . . . . 12
30	29	ralbidv 2986	. . . . . . . . . . 11
31	28, 30	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$
32	27, 31	spcev 3300	. . . . . . . . 9 $/\$ $/\$
33	26, 32	syl6 35	. . . . . . . 8 $/\$ $\/$ $/\$
34	33	exlimdv 1861	. . . . . . 7 $/\$ $\/$ $/\$
35		eleq1 2689	. . . . . . . . 9
36		breq2 4657	. . . . . . . . . 10
37	36	ralbidv 2986	. . . . . . . . 9
38	35, 37	anbi12d 747	. . . . . . . 8 $/\$ $/\$
39	38	cbvexv 2275	. . . . . . 7 $/\$ $/\$
40	34, 39	syl6ib 241	. . . . . 6 $/\$ $\/$ $/\$
41		df-rex 2918	. . . . . 6 $\/$ $/\$ $\/$
42		df-rex 2918	. . . . . 6 $/\$
43	40, 41, 42	3imtr4g 285	. . . . 5 $\/$
44	43	adantr 481	. . . 4 $/\$ $\/$
45	44	imdistani 726	. . 3 $/\$ $/\$ $\/$ $/\$ $/\$
46		df-3an 1039	. . 3 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
47		df-3an 1039	. . 3 $/\$ $/\$ $/\$ $/\$
48	45, 46, 47	3imtr4i 281	. 2 $/\$ $/\$ $\/$ $/\$ $/\$
49		axsup 10113	. 2 $/\$ $/\$ $/\$
50	48, 49	syl 17	1 $/\$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wex 1704

wcel 1990

wne 2794

wral 2912

wrex 2913

wss 3574

c0 3915 class class class wbr 4653 (class class class)co 6650

cr 9935

c1 9937

caddc 9939

clt 10074

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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269

This theorem is referenced by: sup3 10980 nnunb 11288

W3C validator