mnfltpnf - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mnfltpnf		GIF version

Theorem mnfltpnf 9685

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 2157	. . . 4 ⊢ -∞ = -∞
2		eqid 2157	. . . 4 ⊢ +∞ = +∞
3		olc 701	. . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)))
4	1, 2, 3	mp2an 423	. . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))
5	4	orci 721	. 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))
6		mnfxr 7928	. . 3 ⊢ -∞ ∈ ℝ^*
7		pnfxr 7924	. . 3 ⊢ +∞ ∈ ℝ^*
8		ltxr 9675	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))))
9	6, 7, 8	mp2an 423	. 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 698 = wceq 1335 ∈ wcel 2128 class class class wbr 3965 ℝcr 7725 <_ℝ cltrr 7730 +∞cpnf 7903 -∞cmnf 7904 ℝ^*cxr 7905 < clt 7906

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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-cnex 7817

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4591 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911

This theorem is referenced by: mnfltxr 9686 xrlttr 9695 xrltso 9696 xrlttri3 9697 nltpnft 9711 npnflt 9712 ngtmnft 9714 nmnfgt 9715 xltnegi 9732 xposdif 9779

W3C validator