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Theorem mnfltpnf 9571

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 2139	. . . 4 ⊢ -∞ = -∞
2		eqid 2139	. . . 4 ⊢ +∞ = +∞
3		olc 700	. . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)))
4	1, 2, 3	mp2an 422	. . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))
5	4	orci 720	. 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))
6		mnfxr 7822	. . 3 ⊢ -∞ ∈ ℝ^*
7		pnfxr 7818	. . 3 ⊢ +∞ ∈ ℝ^*
8		ltxr 9562	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))))
9	6, 7, 8	mp2an 422	. 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))
10	5, 9	mpbir 145	1 ⊢ -∞ < +∞

Colors of variables: wff set class

Syntax hints: ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 <_ℝ cltrr 7624 +∞cpnf 7797 -∞cmnf 7798 ℝ^*cxr 7799 < clt 7800

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805

This theorem is referenced by: mnfltxr 9572 xrlttr 9581 xrltso 9582 xrlttri3 9583 nltpnft 9597 npnflt 9598 ngtmnft 9600 nmnfgt 9601 xltnegi 9618 xposdif 9665

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