infxrpnf2 - Mathbox for Glauco Siliprandi

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Theorem infxrpnf2 46442

Description: Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Assertion**
Ref	Expression
infxrpnf2	⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

**Proof of Theorem infxrpnf2**
Step	Hyp	Ref	Expression
1		ssdifss 4087	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {+∞}) ⊆ ℝ^*)
2		infxrpnf 46425	. . . . 5 ⊢ ((𝐴 ∖ {+∞}) ⊆ ℝ^* → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
3	1, 2	syl 18	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
4	3	adantr 486	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
5		difsnid 4771	. . . . 5 ⊢ (+∞ ∈ 𝐴 → ((𝐴 ∖ {+∞}) ∪ {+∞}) = 𝐴)
6	5	infeq1d 9463	. . . 4 ⊢ (+∞ ∈ 𝐴 → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
7	6	adantl 487	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
8	4, 7	eqtr3d 2798	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
9		difsn 4761	. . . 4 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {+∞}) = 𝐴)
10	9	infeq1d 9463	. . 3 ⊢ (¬ +∞ ∈ 𝐴 → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
11	10	adantl 487	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ ¬ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
12	8, 11	pm2.61dan 825	1 ⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 {csn 4584 infcinf 9426 +∞cpnf 11333 ℝ^*cxr 11335 < clt 11336

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9427 df-inf 9428 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537

This theorem is used by: supminfxr2 46448

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