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Theorem supxrpnf 12341

Description: The supremum of a set of extended reals containing plus infinity is plus infinity. (Contributed by NM, 15-Oct-2005.)

**Assertion**
Ref	Expression
supxrpnf	⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)

Proof of Theorem supxrpnf Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3738	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*))
2		pnfnlt 12155	. . . . 5 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
3	1, 2	syl6 35	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦))
4	3	ralrimiv 3103	. . 3 ⊢ (𝐴 ⊆ ℝ^* → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
5		breq2 4808	. . . . . 6 ⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞))
6	5	rspcev 3449	. . . . 5 ⊢ ((+∞ ∈ 𝐴 ∧ 𝑦 < +∞) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
7	6	ex 449	. . . 4 ⊢ (+∞ ∈ 𝐴 → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
8	7	ralrimivw 3105	. . 3 ⊢ (+∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
9	4, 8	anim12i 591	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
10		pnfxr 10284	. . 3 ⊢ +∞ ∈ ℝ^*
11		supxr 12336	. . 3 ⊢ (((𝐴 ⊆ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ^, < ) = +∞)
12	10, 11	mpanl2 719	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ^*, < ) = +∞)
13	9, 12	syldan 488	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∃wrex 3051 ⊆ wss 3715 class class class wbr 4804 supcsup 8511 ℝcr 10127 +∞cpnf 10263 ℝ^*cxr 10265 < clt 10266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461

This theorem is referenced by: xrsup 12861 volsup 23524 supxrge 40052 supminfxr2 40197 sge0tsms 41100 sge0sup 41111

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