Theorem lble 9062

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 9060	. . . . 5 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
2		nfcv 2352	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
3		nfriota1 5935	. . . . . . . 8 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
4		nfcv 2352	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
5		nfcv 2352	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
6	3, 4, 5	nfbr 4109	. . . . . . 7 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦
7	2, 6	nfralxy 2548	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦
8		eqid 2209	. . . . . 6 ⊢ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
9		nfra1 2541	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦
10		nfcv 2352	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑆
11	9, 10	nfriota 5937	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
12	11	nfeq2 2364	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
13		breq1 4065	. . . . . . 7 ⊢ (𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (𝑥 ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦))
14	12, 13	ralbid 2508	. . . . . 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 5953	. . . . 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 2352	. . . . 5 ⊢ Ⅎ𝑦 ≤
19		nfcv 2352	. . . . 5 ⊢ Ⅎ𝑦𝐴
20	11, 18, 19	nfbr 4109	. . . 4 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴
21		breq2 4066	. . . 4 ⊢ (𝑦 = 𝐴 → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴))
22	20, 21	rspc 2881	. . 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 1199	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 983   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  ∃!wreu 2490   ⊆ wss 3177   class class class wbr 4062  ℩crio 5926  ℝcr 7966   ≤ cle 8150

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-pre-ltirr 8079  ax-pre-apti 8082

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-xp 4702  df-cnv 4704  df-iota 5254  df-riota 5927  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155

This theorem is referenced by:  lbinf  9063  lbinfle  9065