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Theorem supxrbnd1 13226

Description: The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

**Assertion**
Ref	Expression
supxrbnd1	⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ sup(𝐴, ℝ^*, < ) < +∞))

Distinct variable group: 𝑥,𝑦,𝐴

**Proof of Theorem supxrbnd1**
Step	Hyp	Ref	Expression
1		ralnex 3058	. . . 4 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥)
2		rexr 11164	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
3		ssel2 3924	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
4		xrlenlt 11183	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
5	2, 3, 4	syl2anr 597	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
6	5	an32s 652	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
7	6	rexbidva 3154	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥))
8		rexnal 3084	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥)
9	7, 8	bitr2di 288	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
10	9	ralbidva 3153	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
11	1, 10	bitr3id 285	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
12		supxrunb1 13224	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
13		supxrcl 13220	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
14		nltpnft 13069	. . . 4 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
15	13, 14	syl 17	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
16	11, 12, 15	3bitrd 305	. 2 ⊢ (𝐴 ⊆ ℝ^* → (¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
17	16	con4bid 317	1 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ sup(𝐴, ℝ^*, < ) < +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ⊆ wss 3897 class class class wbr 5093 supcsup 9330 ℝcr 11011 +∞cpnf 11149 ℝ^*cxr 11151 < clt 11152 ≤ cle 11153

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9332 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353

This theorem is referenced by: (None)

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