supxrre2 - Metamath Proof Explorer

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Theorem supxrre2 13375

Description: The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)

**Assertion**
Ref	Expression
supxrre2	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))

**Proof of Theorem supxrre2**
Step	Hyp	Ref	Expression
1		supxrre1 13374	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) < +∞))
2		ressxr 11271	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
3		sstr 3948	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
4	2, 3	mpan2 704	. . . . 5 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
5		supxrcl 13359	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
6		nltpnft 13208	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
7	4, 5, 6	3syl 19	. . . 4 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
8	7	necon2abid 3003	. . 3 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ^, < ) < +∞ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))
9	8	adantr 486	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) < +∞ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))
10	1, 9	bitrd 282	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ⊆ wss 3908 ∅c0 4289 class class class wbr 5114 supcsup 9410 ℝcr 11117 +∞cpnf 11258 ℝ^*cxr 11260 < clt 11261

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-po 5574 df-so 5575 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462

This theorem is used by: ovollb2 25685 nmorepnf 31157 nmoprepnf 32256 nmfnrepnf 32269

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