Theorem ublbneg 9432

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 9417. (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 3940	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑏 ≤ 𝑎 ↔ 𝑦 ≤ 𝑎))
2	1	cbvralv 2657	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
3	2	rexbii 2445	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
4		breq2 3941	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑦 ≤ 𝑎 ↔ 𝑦 ≤ 𝑥))
5	4	ralbidv 2438	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
6	5	cbvrexv 2658	. . 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 8047	. . . 4 ⊢ (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
9		elrabi 2841	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → 𝑦 ∈ ℝ)
10		negeq 7979	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → -𝑧 = -𝑦)
11	10	eleq1d 2209	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-𝑧 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
12	11	elrab3 2845	. . . . . . . . . 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 3940	. . . . . . . . 9 ⊢ (𝑏 = -𝑦 → (𝑏 ≤ 𝑎 ↔ -𝑦 ≤ 𝑎))
16	15	rspcv 2789	. . . . . . . 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 8252	. . . . . . 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 2515	. . . 4 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
23		breq1 3940	. . . . . 6 ⊢ (𝑥 = -𝑎 → (𝑥 ≤ 𝑦 ↔ -𝑎 ≤ 𝑦))
24	23	ralbidv 2438	. . . . 5 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
25	24	rspcev 2793	. . . 4 ⊢ ((-𝑎 ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
26	8, 22, 25	syl6an 1411	. . 3 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦))
27	26	rexlimiv 2546	. 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 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {crab 2421   class class class wbr 3937  ℝcr 7643   ≤ cle 7825  -cneg 7958

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960

This theorem is referenced by: (None)