supxrmnf - Metamath Proof Explorer

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Theorem supxrmnf 13356

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 4168	. . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴)
2	1	supeq1i 9485	. 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(({-∞} ∪ 𝐴), ℝ^, < )
3		mnfxr 11316	. . . 4 ⊢ -∞ ∈ ℝ^*
4		snssi 4813	. . . 4 ⊢ (-∞ ∈ ℝ^* → {-∞} ⊆ ℝ^*)
5	3, 4	mp1i 13	. . 3 ⊢ (𝐴 ⊆ ℝ^* → {-∞} ⊆ ℝ^*)
6		id 22	. . 3 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ⊆ ℝ^*)
7		xrltso 13180	. . . . 5 ⊢ < Or ℝ^*
8		supsn 9510	. . . . 5 ⊢ (( < Or ℝ^* ∧ -∞ ∈ ℝ^) → sup({-∞}, ℝ^, < ) = -∞)
9	7, 3, 8	mp2an 692	. . . 4 ⊢ sup({-∞}, ℝ^*, < ) = -∞
10		supxrcl 13354	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
11		mnfle 13174	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐴, ℝ^*, < ))
12	10, 11	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → -∞ ≤ sup(𝐴, ℝ^*, < ))
13	9, 12	eqbrtrid 5183	. . 3 ⊢ (𝐴 ⊆ ℝ^* → sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < ))
14		supxrun 13355	. . 3 ⊢ (({-∞} ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^* ∧ sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < )) → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
15	5, 6, 13, 14	syl3anc 1370	. 2 ⊢ (𝐴 ⊆ ℝ^* → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
16	2, 15	eqtrid 2787	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 class class class wbr 5148 Or wor 5596 supcsup 9478 -∞cmnf 11291 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493

This theorem is referenced by: supxrmnf2 45383

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