Theorem ublbneg 9734

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 9716. (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 4047	. . . . 5 ⊢ (𝑏 = 𝑦 → (𝑏 ≤ 𝑎 ↔ 𝑦 ≤ 𝑎))
2	1	cbvralv 2738	. . . 4 ⊢ (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
3	2	rexbii 2513	. . 3 ⊢ (∃𝑎 ∈ ℝ ∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃𝑎 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎)
4		breq2 4048	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑦 ≤ 𝑎 ↔ 𝑦 ≤ 𝑥))
5	4	ralbidv 2506	. . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
6	5	cbvrexv 2739	. . 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 8333	. . . 4 ⊢ (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
9		elrabi 2926	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} → 𝑦 ∈ ℝ)
10		negeq 8265	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → -𝑧 = -𝑦)
11	10	eleq1d 2274	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (-𝑧 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
12	11	elrab3 2930	. . . . . . . . . 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 4047	. . . . . . . . 9 ⊢ (𝑏 = -𝑦 → (𝑏 ≤ 𝑎 ↔ -𝑦 ≤ 𝑎))
16	15	rspcv 2873	. . . . . . . 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 8539	. . . . . . 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 2586	. . . 4 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
23		breq1 4047	. . . . . 6 ⊢ (𝑥 = -𝑎 → (𝑥 ≤ 𝑦 ↔ -𝑎 ≤ 𝑦))
24	23	ralbidv 2506	. . . . 5 ⊢ (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦))
25	24	rspcev 2877	. . . 4 ⊢ ((-𝑎 ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}-𝑎 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
26	8, 22, 25	syl6an 1454	. . 3 ⊢ (𝑎 ∈ ℝ → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦))
27	26	rexlimiv 2617	. 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 1373   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  {crab 2488   class class class wbr 4044  ℝcr 7924   ≤ cle 8108  -cneg 8244

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246

This theorem is referenced by: (None)