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Theorem supxrbnd1 13461

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 3091	. . . 4 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥)
2		rexr 11365	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
3		ssel2 3907	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
4		xrlenlt 11384	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
5	2, 3, 4	syl2anr 609	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
6	5	an32s 665	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
7	6	rexbidva 3187	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥))
8		rexnal 3117	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥)
9	7, 8	bitr2di 291	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
10	9	ralbidva 3186	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
11	1, 10	bitr3id 288	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
12		supxrunb1 13459	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
13		supxrcl 13455	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
14		nltpnft 13304	. . . 4 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
15	13, 14	syl 18	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
16	11, 12, 15	3bitrd 308	. 2 ⊢ (𝐴 ⊆ ℝ^* → (¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
17	16	con4bid 320	1 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ sup(𝐴, ℝ^*, < ) < +∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ∀wral 3079 ∃wrex 3089 ⊆ wss 3899 class class class wbr 5105 supcsup 9442 ℝcr 11209 +∞cpnf 11350 ℝ^*cxr 11352 < clt 11353 ≤ cle 11354

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-br 5106 df-opab 5170 df-mpt 5189 df-id 5549 df-po 5562 df-so 5563 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9444 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554

This theorem is used by: (None)

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