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Theorem ublbneg 9067
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 9052. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
ublbneg (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem ublbneg
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3840 . . . . 5 (𝑏 = 𝑦 → (𝑏𝑎𝑦𝑎))
21cbvralv 2590 . . . 4 (∀𝑏𝐴 𝑏𝑎 ↔ ∀𝑦𝐴 𝑦𝑎)
32rexbii 2385 . . 3 (∃𝑎 ∈ ℝ ∀𝑏𝐴 𝑏𝑎 ↔ ∃𝑎 ∈ ℝ ∀𝑦𝐴 𝑦𝑎)
4 breq2 3841 . . . . 5 (𝑎 = 𝑥 → (𝑦𝑎𝑦𝑥))
54ralbidv 2380 . . . 4 (𝑎 = 𝑥 → (∀𝑦𝐴 𝑦𝑎 ↔ ∀𝑦𝐴 𝑦𝑥))
65cbvrexv 2591 . . 3 (∃𝑎 ∈ ℝ ∀𝑦𝐴 𝑦𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)
73, 6bitri 182 . 2 (∃𝑎 ∈ ℝ ∀𝑏𝐴 𝑏𝑎 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)
8 renegcl 7722 . . . 4 (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
9 elrabi 2766 . . . . . . . . 9 (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴} → 𝑦 ∈ ℝ)
10 negeq 7654 . . . . . . . . . . . 12 (𝑧 = 𝑦 → -𝑧 = -𝑦)
1110eleq1d 2156 . . . . . . . . . . 11 (𝑧 = 𝑦 → (-𝑧𝐴 ↔ -𝑦𝐴))
1211elrab3 2770 . . . . . . . . . 10 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴} ↔ -𝑦𝐴))
1312biimpd 142 . . . . . . . . 9 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴} → -𝑦𝐴))
149, 13mpcom 36 . . . . . . . 8 (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴} → -𝑦𝐴)
15 breq1 3840 . . . . . . . . 9 (𝑏 = -𝑦 → (𝑏𝑎 ↔ -𝑦𝑎))
1615rspcv 2718 . . . . . . . 8 (-𝑦𝐴 → (∀𝑏𝐴 𝑏𝑎 → -𝑦𝑎))
1714, 16syl 14 . . . . . . 7 (𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴} → (∀𝑏𝐴 𝑏𝑎 → -𝑦𝑎))
1817adantl 271 . . . . . 6 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}) → (∀𝑏𝐴 𝑏𝑎 → -𝑦𝑎))
19 lenegcon1 7923 . . . . . . 7 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎𝑦 ↔ -𝑦𝑎))
209, 19sylan2 280 . . . . . 6 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}) → (-𝑎𝑦 ↔ -𝑦𝑎))
2118, 20sylibrd 167 . . . . 5 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}) → (∀𝑏𝐴 𝑏𝑎 → -𝑎𝑦))
2221ralrimdva 2453 . . . 4 (𝑎 ∈ ℝ → (∀𝑏𝐴 𝑏𝑎 → ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}-𝑎𝑦))
23 breq1 3840 . . . . . 6 (𝑥 = -𝑎 → (𝑥𝑦 ↔ -𝑎𝑦))
2423ralbidv 2380 . . . . 5 (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦 ↔ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}-𝑎𝑦))
2524rspcev 2722 . . . 4 ((-𝑎 ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}-𝑎𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)
268, 22, 25syl6an 1368 . . 3 (𝑎 ∈ ℝ → (∀𝑏𝐴 𝑏𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦))
2726rexlimiv 2483 . 2 (∃𝑎 ∈ ℝ ∀𝑏𝐴 𝑏𝑎 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)
287, 27sylbir 133 1 (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  wrex 2360  {crab 2363   class class class wbr 3837  cr 7328  cle 7502  -cneg 7633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635
This theorem is referenced by: (None)
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