elxrge0 - Intuitionistic Logic Explorer

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Theorem elxrge0 10047

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 8068	. . 3 ⊢ 0 ∈ ℝ^*
3		pnfxr 8074	. . 3 ⊢ +∞ ∈ ℝ^*
4		elicc1 9993	. . 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 9858	. . . 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 2164 class class class wbr 4030 (class class class)co 5919 0cc0 7874 +∞cpnf 8053 ℝ^*cxr 8055 ≤ cle 8057 [,]cicc 9960

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 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1re 7968 ax-addrcl 7971 ax-rnegex 7983

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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-icc 9964

This theorem is referenced by: 0e0iccpnf 10049 ge0xaddcl 10052 psmetxrge0 14511 isxmet2d 14527 comet 14678 bdxmet 14680

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