Theorem ublbneg 9627

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 9609. (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 4018	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑏 ≤ 𝑎 ↔ 𝑦 ≤ 𝑎))
2	1	cbvralv 2715	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
3	2	rexbii 2494	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
4		breq2 4019	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑦 ≤ 𝑎 ↔ 𝑦 ≤ 𝑥))
5	4	ralbidv 2487	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
6	5	cbvrexv 2716	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
7	3, 6	bitri 184	. 2 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
8		renegcl 8232	. . . 4 ⊢ (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
9		elrabi 2902	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → 𝑦 ∈ ℝ)
10		negeq 8164	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → -𝑧 = -𝑦)
11	10	eleq1d 2256	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-𝑧 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
12	11	elrab3 2906	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ↔ -𝑦 ∈ 𝐴))
13	12	biimpd 144	. . . . . . . . 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 4018	. . . . . . . . 9 ⊢ (𝑏 = -𝑦 → (𝑏 ≤ 𝑎 ↔ -𝑦 ≤ 𝑎))
16	15	rspcv 2849	. . . . . . . 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 277	. . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → -𝑦 ≤ 𝑎))
19		lenegcon1 8437	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎 ≤ 𝑦 ↔ -𝑦 ≤ 𝑎))
20	9, 19	sylan2 286	. . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) → (-𝑎 ≤ 𝑦 ↔ -𝑦 ≤ 𝑎))
21	18, 20	sylibrd 169	. . . . 5 ⊢ ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → -𝑎 ≤ 𝑦))
22	21	ralrimdva 2567	. . . 4 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
23		breq1 4018	. . . . . 6 ⊢ (𝑥 = -𝑎 → (𝑥 ≤ 𝑦 ↔ -𝑎 ≤ 𝑦))
24	23	ralbidv 2487	. . . . 5 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
25	24	rspcev 2853	. . . 4 ⊢ ((-𝑎 ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
26	8, 22, 25	syl6an 1444	. . 3 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦))
27	26	rexlimiv 2598	. 2 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
28	7, 27	sylbir 135	1 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363   ∈ wcel 2158  ∀wral 2465  ∃wrex 2466  {crab 2469   class class class wbr 4015  ℝcr 7824   ≤ cle 8007  -cneg 8143

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145

This theorem is referenced by: (None)