xrsupssd - Mathbox for Thierry Arnoux

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Theorem xrsupssd 29524

Description: Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)

**Hypotheses**
Ref	Expression
xrsupssd.1	⊢ (𝜑 → 𝐵 ⊆ 𝐶)
xrsupssd.2	⊢ (𝜑 → 𝐶 ⊆ ℝ^*)

**Assertion**
Ref	Expression
xrsupssd	⊢ (𝜑 → sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ))

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

Step	Hyp	Ref	Expression
1		xrltso 11974	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		xrsupssd.1	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4		xrsupssd.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ ℝ^*)
5	3, 4	sstrd 3613	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
6		xrsupss 12139	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))
8		xrsupss 12139	. . . 4 ⊢ (𝐶 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐶 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐶 𝑦 < 𝑧)))
9	4, 8	syl 17	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐶 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐶 𝑦 < 𝑧)))
10	2, 3, 4, 7, 9	supssd 29487	. 2 ⊢ (𝜑 → ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < ))
11	2, 7	supcl 8364	. . 3 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
12	2, 9	supcl 8364	. . 3 ⊢ (𝜑 → sup(𝐶, ℝ^, < ) ∈ ℝ^)
13		xrlenlt 10103	. . 3 ⊢ ((sup(𝐵, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐶, ℝ^, < ) ∈ ℝ^) → (sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ) ↔ ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < )))
14	11, 12, 13	syl2anc 693	. 2 ⊢ (𝜑 → (sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ) ↔ ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < )))
15	10, 14	mpbird 247	1 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 Or wor 5034 supcsup 8346 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075

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 ax-pre-sup 10014

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: (None)

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