xrsupssd - Mathbox for Thierry Arnoux

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Theorem xrsupssd 32663

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 13174	. . . 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 3990	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
6		xrsupss 13342	. . . 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 13342	. . . 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 32623	. 2 ⊢ (𝜑 → ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < ))
11	2, 7	supcl 9501	. . 3 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
12	2, 9	supcl 9501	. . 3 ⊢ (𝜑 → sup(𝐶, ℝ^, < ) ∈ ℝ^)
13		xrlenlt 11329	. . 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 582	. 2 ⊢ (𝜑 → (sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ) ↔ ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < )))
15	10, 14	mpbird 256	1 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ⊆ wss 3947 class class class wbr 5153 Or wor 5593 supcsup 9483 ℝ^*cxr 11297 < clt 11298 ≤ cle 11299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497

This theorem is referenced by: (None)

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