xrred - Mathbox for Glauco Siliprandi

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Theorem xrred 46581

Description: An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
xrred.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrred.2	⊢ (𝜑 → 𝐴 ≠ -∞)
xrred.3	⊢ (𝜑 → 𝐴 ≠ +∞)

**Assertion**
Ref	Expression
xrred	⊢ (𝜑 → 𝐴 ∈ ℝ)

**Proof of Theorem xrred**
Step	Hyp	Ref	Expression
1		xrred.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xrred.2	. . . 4 ⊢ (𝜑 → 𝐴 ≠ -∞)
3	1, 2	jca 521	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞))
4		xrnemnf 13273	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
5	3, 4	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
6		xrred.3	. . 3 ⊢ (𝜑 → 𝐴 ≠ +∞)
7	6	neneqd 2964	. 2 ⊢ (𝜑 → ¬ 𝐴 = +∞)
8		pm2.53 865	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞))
9	8	con1d 146	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 = +∞ → 𝐴 ∈ ℝ))
10	5, 7, 9	sylc 66	1 ⊢ (𝜑 → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ℝcr 11226 +∞cpnf 11367 -∞cmnf 11368 ℝ^*cxr 11369

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-pnf 11372 df-mnf 11373 df-xr 11374

This theorem is used by: infxr 46583 infleinflem2 46587 xrralrecnnge 46606 xrre4 46626 supminfxr2 46684 xrpnf 46700 climxrrelem 46964 climxrre 46965 liminflimsupxrre 47032 ioorrnopnxrlem 47521 pimiooltgt 47925 smfpimltxr 47962 smfpimgtxr 47995

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