supxrpnf - Metamath Proof Explorer

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Theorem supxrpnf 12148

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 3597	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*))
2		pnfnlt 11962	. . . . 5 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
3	1, 2	syl6 35	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦))
4	3	ralrimiv 2965	. . 3 ⊢ (𝐴 ⊆ ℝ^* → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
5		breq2 4657	. . . . . 6 ⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞))
6	5	rspcev 3309	. . . . 5 ⊢ ((+∞ ∈ 𝐴 ∧ 𝑦 < +∞) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
7	6	ex 450	. . . 4 ⊢ (+∞ ∈ 𝐴 → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
8	7	ralrimivw 2967	. . 3 ⊢ (+∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
9	4, 8	anim12i 590	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
10		pnfxr 10092	. . 3 ⊢ +∞ ∈ ℝ^*
11		supxr 12143	. . 3 ⊢ (((𝐴 ⊆ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ^, < ) = +∞)
12	10, 11	mpanl2 717	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ^*, < ) = +∞)
13	9, 12	syldan 487	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ^*, < ) = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 supcsup 8346 ℝcr 9935 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269

This theorem is referenced by: xrsup 12667 volsup 23324 supxrge 39554 supminfxr2 39699 sge0tsms 40597 sge0sup 40608

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