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Theorem mnfltpnf 11960

Description: Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnfltpnf	⊢ -∞ < +∞

**Proof of Theorem mnfltpnf**
Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ -∞ = -∞
2		eqid 2622	. . . 4 ⊢ +∞ = +∞
3		olc 399	. . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)))
4	1, 2, 3	mp2an 708	. . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))
5	4	orci 405	. 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))
6		mnfxr 10096	. . 3 ⊢ -∞ ∈ ℝ^*
7		pnfxr 10092	. . 3 ⊢ +∞ ∈ ℝ^*
8		ltxr 11949	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))))
9	6, 7, 8	mp2an 708	. 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))
10	5, 9	mpbir 221	1 ⊢ -∞ < +∞

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 <_ℝ cltrr 9940 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < 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-cnex 9992

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-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-xp 5120 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079

This theorem is referenced by: mnfltxr 11961 xrlttri 11972 xrlttr 11973 xltnegi 12047 supxrltinfxr 39677 liminflelimsupcex 40029

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