Theorem ublbneg 9405

Description: The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9390. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
ublbneg	⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem ublbneg

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 3932	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑏 ≤ 𝑎 ↔ 𝑦 ≤ 𝑎))
2	1	cbvralv 2654	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
3	2	rexbii 2442	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
4		breq2 3933	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑦 ≤ 𝑎 ↔ 𝑦 ≤ 𝑥))
5	4	ralbidv 2437	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
6	5	cbvrexv 2655	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
7	3, 6	bitri 183	. 2 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
8		renegcl 8023	. . . 4 ⊢ (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
9		elrabi 2837	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → 𝑦 ∈ ℝ)
10		negeq 7955	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → -𝑧 = -𝑦)
11	10	eleq1d 2208	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-𝑧 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
12	11	elrab3 2841	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ -𝑦 ∈ 𝐴))
13	12	biimpd 143	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → -𝑦 ∈ 𝐴))
14	9, 13	mpcom 36	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → -𝑦 ∈ 𝐴)
15		breq1 3932	. . . . . . . . 9 ⊢ (𝑏 = -𝑦 → (𝑏 ≤ 𝑎 ↔ -𝑦 ≤ 𝑎))
16	15	rspcv 2785	. . . . . . . 8 ⊢ (-𝑦 ∈ 𝐴 → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → -𝑦 ≤ 𝑎))
17	14, 16	syl 14	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → -𝑦 ≤ 𝑎))
18	17	adantl 275	. . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → -𝑦 ≤ 𝑎))
19		lenegcon1 8228	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎 ≤ 𝑦 ↔ -𝑦 ≤ 𝑎))
20	9, 19	sylan2 284	. . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) → (-𝑎 ≤ 𝑦 ↔ -𝑦 ≤ 𝑎))
21	18, 20	sylibrd 168	. . . . 5 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → -𝑎 ≤ 𝑦))
22	21	ralrimdva 2512	. . . 4 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
23		breq1 3932	. . . . . 6 ⊢ (𝑥 = -𝑎 → (𝑥 ≤ 𝑦 ↔ -𝑎 ≤ 𝑦))
24	23	ralbidv 2437	. . . . 5 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
25	24	rspcev 2789	. . . 4 ⊢ ((-𝑎 ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
26	8, 22, 25	syl6an 1410	. . 3 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦))
27	26	rexlimiv 2543	. 2 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
28	7, 27	sylbir 134	1 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   class class class wbr 3929  ℝcr 7619   ≤ cle 7801  -cneg 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936

This theorem is referenced by: (None)