Theorem lble 10854

Description: If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
lble	⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)

Distinct variable groups:   𝑥,𝑦,𝑆   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem lble

Step	Hyp	Ref	Expression
1		lbreu 10852	. . . . 5 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
2		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
3		nfriota1 6518	. . . . . . . 8 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
4		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
6	3, 4, 5	nfbr 4629	. . . . . . 7 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦
7	2, 6	nfral 2929	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦
8		eqid 2610	. . . . . 6 ⊢ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
9		nfra1 2925	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦
10		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑆
11	9, 10	nfriota 6520	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
12	11	nfeq2 2766	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
13		breq1 4586	. . . . . . 7 ⊢ (𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (𝑥 ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦))
14	12, 13	ralbid 2966	. . . . . 6 ⊢ (𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦))
15	7, 8, 14	riotaprop 6534	. . . . 5 ⊢ (∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦))
16	1, 15	syl 17	. . . 4 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦))
17	16	simprd 478	. . 3 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦)
18		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑦 ≤
19		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑦𝐴
20	11, 18, 19	nfbr 4629	. . . 4 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴
21		breq2 4587	. . . 4 ⊢ (𝑦 = 𝐴 → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴))
22	20, 21	rspc 3276	. . 3 ⊢ (𝐴 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴))
23	17, 22	mpan9 485	. 2 ⊢ (((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)
24	23	3impa 1251	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ⊆ wss 3540   class class class wbr 4583  ℩crio 6510  ℝcr 9814   ≤ cle 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959

This theorem is referenced by:  lbinf  10855  lbinfle  10857