supxrmnf - Metamath Proof Explorer

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Theorem supxrmnf 13232

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 4110	. . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴)
2	1	supeq1i 9350	. 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(({-∞} ∪ 𝐴), ℝ^, < )
3		mnfxr 11189	. . . 4 ⊢ -∞ ∈ ℝ^*
4		snssi 4764	. . . 4 ⊢ (-∞ ∈ ℝ^* → {-∞} ⊆ ℝ^*)
5	3, 4	mp1i 13	. . 3 ⊢ (𝐴 ⊆ ℝ^* → {-∞} ⊆ ℝ^*)
6		id 22	. . 3 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ⊆ ℝ^*)
7		xrltso 13055	. . . . 5 ⊢ < Or ℝ^*
8		supsn 9376	. . . . 5 ⊢ (( < Or ℝ^* ∧ -∞ ∈ ℝ^) → sup({-∞}, ℝ^, < ) = -∞)
9	7, 3, 8	mp2an 692	. . . 4 ⊢ sup({-∞}, ℝ^*, < ) = -∞
10		supxrcl 13230	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
11		mnfle 13049	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐴, ℝ^*, < ))
12	10, 11	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → -∞ ≤ sup(𝐴, ℝ^*, < ))
13	9, 12	eqbrtrid 5133	. . 3 ⊢ (𝐴 ⊆ ℝ^* → sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < ))
14		supxrun 13231	. . 3 ⊢ (({-∞} ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^* ∧ sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < )) → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
15	5, 6, 13, 14	syl3anc 1373	. 2 ⊢ (𝐴 ⊆ ℝ^* → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
16	2, 15	eqtrid 2783	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ∪ cun 3899 ⊆ wss 3901 {csn 4580 class class class wbr 5098 Or wor 5531 supcsup 9343 -∞cmnf 11164 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367

This theorem is referenced by: supxrmnf2 45677

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