Theorem lble 9102

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 9100	. . . . 5 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
2		nfcv 2372	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
3		nfriota1 5968	. . . . . . . 8 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
4		nfcv 2372	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2372	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
6	3, 4, 5	nfbr 4130	. . . . . . 7 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦
7	2, 6	nfralxy 2568	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦
8		eqid 2229	. . . . . 6 ⊢ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
9		nfra1 2561	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦
10		nfcv 2372	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑆
11	9, 10	nfriota 5970	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
12	11	nfeq2 2384	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
13		breq1 4086	. . . . . . 7 ⊢ (𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (𝑥 ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦))
14	12, 13	ralbid 2528	. . . . . 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 5986	. . . . 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 14	. . . 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 114	. . 3 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦)
18		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑦 ≤
19		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑦𝐴
20	11, 18, 19	nfbr 4130	. . . 4 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴
21		breq2 4087	. . . 4 ⊢ (𝑦 = 𝐴 → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴))
22	20, 21	rspc 2901	. . 3 ⊢ (𝐴 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴))
23	17, 22	mpan9 281	. 2 ⊢ (((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)
24	23	3impa 1218	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510   ⊆ wss 3197   class class class wbr 4083  ℩crio 5959  ℝcr 8006   ≤ cle 8190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-pre-ltirr 8119  ax-pre-apti 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-iota 5278  df-riota 5960  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195

This theorem is referenced by:  lbinf  9103  lbinfle  9105