Theorem nltpnft 9966

Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Assertion

Ref	Expression
nltpnft	|- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Proof of Theorem nltpnft

Step	Hyp	Ref	Expression
1		elxr 9928	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renepnf 8150	. . . . 5 |- ( A e. RR -> A =/= +oo )
3	2	neneqd 2398	. . . 4 |- ( A e. RR -> -. A = +oo )
4		ltpnf 9932	. . . . 5 |- ( A e. RR -> A < +oo )
5		notnot 630	. . . . 5 |- ( A < +oo -> -. -. A < +oo )
6	4, 5	syl 14	. . . 4 |- ( A e. RR -> -. -. A < +oo )
7	3, 6	2falsed 704	. . 3 |- ( A e. RR -> ( A = +oo <-> -. A < +oo ) )
8		id 19	. . . 4 |- ( A = +oo -> A = +oo )
9		pnfxr 8155	. . . . . 6 |- +oo e. RR*
10		xrltnr 9931	. . . . . 6 |- ( +oo e. RR* -> -. +oo < +oo )
11	9, 10	ax-mp 5	. . . . 5 |- -. +oo < +oo
12		breq1 4057	. . . . 5 |- ( A = +oo -> ( A < +oo <-> +oo < +oo ) )
13	11, 12	mtbiri 677	. . . 4 |- ( A = +oo -> -. A < +oo )
14	8, 13	2thd 175	. . 3 |- ( A = +oo -> ( A = +oo <-> -. A < +oo ) )
15		mnfnepnf 8158	. . . . . 6 |- -oo =/= +oo
16	15	neii 2379	. . . . 5 |- -. -oo = +oo
17		eqeq1 2213	. . . . 5 |- ( A = -oo -> ( A = +oo <-> -oo = +oo ) )
18	16, 17	mtbiri 677	. . . 4 |- ( A = -oo -> -. A = +oo )
19		mnfltpnf 9937	. . . . . . 7 |- -oo < +oo
20		breq1 4057	. . . . . . 7 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
21	19, 20	mpbiri 168	. . . . . 6 |- ( A = -oo -> A < +oo )
22	21	necon3bi 2427	. . . . 5 |- ( -. A < +oo -> A =/= -oo )
23	22	necon2bi 2432	. . . 4 |- ( A = -oo -> -. -. A < +oo )
24	18, 23	2falsed 704	. . 3 |- ( A = -oo -> ( A = +oo <-> -. A < +oo ) )
25	7, 14, 24	3jaoi 1316	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A = +oo <-> -. A < +oo ) )
26	1, 25	sylbi 121	1 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ w3o 980   = wceq 1373   e. wcel 2177   class class class wbr 4054   RRcr 7954   +oocpnf 8134   -oocmnf 8135   RR*cxr 8136   < clt 8137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-pre-ltirr 8067

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-xp 4694  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142

This theorem is referenced by:  npnflt  9967  xgepnf  9968  xrmaxiflemlub  11644