Theorem ublbneg 9963

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 9945. (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 4117	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑏 ≤ 𝑎 ↔ 𝑦 ≤ 𝑎))
2	1	cbvralv 2780	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
3	2	rexbii 2551	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
4		breq2 4118	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑦 ≤ 𝑎 ↔ 𝑦 ≤ 𝑥))
5	4	ralbidv 2544	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
6	5	cbvrexv 2781	. . 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 8550	. . . 4 ⊢ (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
9		elrabi 2973	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → 𝑦 ∈ ℝ)
10		negeq 8482	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → -𝑧 = -𝑦)
11	10	eleq1d 2303	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-𝑧 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
12	11	elrab3 2977	. . . . . . . . . 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 4117	. . . . . . . . 9 ⊢ (𝑏 = -𝑦 → (𝑏 ≤ 𝑎 ↔ -𝑦 ≤ 𝑎))
16	15	rspcv 2919	. . . . . . . 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 8757	. . . . . . 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 2624	. . . 4 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
23		breq1 4117	. . . . . 6 ⊢ (𝑥 = -𝑎 → (𝑥 ≤ 𝑦 ↔ -𝑎 ≤ 𝑦))
24	23	ralbidv 2544	. . . . 5 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
25	24	rspcev 2923	. . . 4 ⊢ ((-𝑎 ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
26	8, 22, 25	syl6an 1479	. . 3 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦))
27	26	rexlimiv 2656	. 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 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526   class class class wbr 4114  ℝcr 8142   ≤ cle 8325  -cneg 8461

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463

This theorem is referenced by: (None)