supxrmnf - Metamath Proof Explorer

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Theorem supxrmnf 13360

Description: Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
supxrmnf	⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

**Proof of Theorem supxrmnf**
Step	Hyp	Ref	Expression
1		uncom 4157	. . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴)
2	1	supeq1i 9488	. 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(({-∞} ∪ 𝐴), ℝ^, < )
3		mnfxr 11319	. . . 4 ⊢ -∞ ∈ ℝ^*
4		snssi 4807	. . . 4 ⊢ (-∞ ∈ ℝ^* → {-∞} ⊆ ℝ^*)
5	3, 4	mp1i 13	. . 3 ⊢ (𝐴 ⊆ ℝ^* → {-∞} ⊆ ℝ^*)
6		id 22	. . 3 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ⊆ ℝ^*)
7		xrltso 13184	. . . . 5 ⊢ < Or ℝ^*
8		supsn 9513	. . . . 5 ⊢ (( < Or ℝ^* ∧ -∞ ∈ ℝ^) → sup({-∞}, ℝ^, < ) = -∞)
9	7, 3, 8	mp2an 692	. . . 4 ⊢ sup({-∞}, ℝ^*, < ) = -∞
10		supxrcl 13358	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
11		mnfle 13178	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐴, ℝ^*, < ))
12	10, 11	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → -∞ ≤ sup(𝐴, ℝ^*, < ))
13	9, 12	eqbrtrid 5177	. . 3 ⊢ (𝐴 ⊆ ℝ^* → sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < ))
14		supxrun 13359	. . 3 ⊢ (({-∞} ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^* ∧ sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < )) → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
15	5, 6, 13, 14	syl3anc 1372	. 2 ⊢ (𝐴 ⊆ ℝ^* → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
16	2, 15	eqtrid 2788	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 {csn 4625 class class class wbr 5142 Or wor 5590 supcsup 9481 -∞cmnf 11294 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496

This theorem is referenced by: supxrmnf2 45449

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