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Theorem supxrpnf 12787

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 3868	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*))
2		pnfnlt 12599	. . . . 5 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
3	1, 2	syl6 35	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦))
4	3	ralrimiv 3095	. . 3 ⊢ (𝐴 ⊆ ℝ^* → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
5		breq2 5031	. . . . . 6 ⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞))
6	5	rspcev 3524	. . . . 5 ⊢ ((+∞ ∈ 𝐴 ∧ 𝑦 < +∞) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
7	6	ex 416	. . . 4 ⊢ (+∞ ∈ 𝐴 → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
8	7	ralrimivw 3097	. . 3 ⊢ (+∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
9	4, 8	anim12i 616	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
10		pnfxr 10766	. . 3 ⊢ +∞ ∈ ℝ^*
11		supxr 12782	. . 3 ⊢ (((𝐴 ⊆ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ^, < ) = +∞)
12	10, 11	mpanl2 701	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ^*, < ) = +∞)
13	9, 12	syldan 594	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ⊆ wss 3841 class class class wbr 5027 supcsup 8970 ℝcr 10607 +∞cpnf 10743 ℝ^*cxr 10745 < clt 10746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-sup 8972 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944

This theorem is referenced by: xrsup 13320 volsup 24301 supxrge 42399 supminfxr2 42533 sge0tsms 43444 sge0sup 43455

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