elxrge0 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > elxrge0		GIF version

Theorem elxrge0 9996

Description: Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

**Assertion**
Ref	Expression
elxrge0	⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴))

**Proof of Theorem elxrge0**
Step	Hyp	Ref	Expression
1		df-3an 982	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
2		0xr 8022	. . 3 ⊢ 0 ∈ ℝ^*
3		pnfxr 8028	. . 3 ⊢ +∞ ∈ ℝ^*
4		elicc1 9942	. . 3 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞)))
5	2, 3, 4	mp2an 426	. 2 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞))
6		pnfge 9807	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7	6	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≤ +∞)
8	7	pm4.71i 391	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
9	1, 5, 8	3bitr4i 212	1 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 980 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5891 0cc0 7829 +∞cpnf 8007 ℝ^*cxr 8009 ≤ cle 8011 [,]cicc 9909

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 ax-rnegex 7938

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-icc 9913

This theorem is referenced by: 0e0iccpnf 9998 ge0xaddcl 10001 psmetxrge0 14229 isxmet2d 14245 comet 14396 bdxmet 14398

W3C validator