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Theorem lbreu 8901
Description: If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
Assertion
Ref Expression
lbreu ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → ∃!𝑥𝑆𝑦𝑆 𝑥𝑦)
Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem lbreu
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 breq2 4007 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑥𝑦𝑥𝑤))
21rspcv 2837 . . . . . . . 8 (𝑤𝑆 → (∀𝑦𝑆 𝑥𝑦𝑥𝑤))
3 breq2 4007 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑤𝑦𝑤𝑥))
43rspcv 2837 . . . . . . . 8 (𝑥𝑆 → (∀𝑦𝑆 𝑤𝑦𝑤𝑥))
52, 4im2anan9r 599 . . . . . . 7 ((𝑥𝑆𝑤𝑆) → ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → (𝑥𝑤𝑤𝑥)))
6 ssel 3149 . . . . . . . . . . . 12 (𝑆 ⊆ ℝ → (𝑥𝑆𝑥 ∈ ℝ))
7 ssel 3149 . . . . . . . . . . . 12 (𝑆 ⊆ ℝ → (𝑤𝑆𝑤 ∈ ℝ))
86, 7anim12d 335 . . . . . . . . . . 11 (𝑆 ⊆ ℝ → ((𝑥𝑆𝑤𝑆) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ)))
98impcom 125 . . . . . . . . . 10 (((𝑥𝑆𝑤𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ))
10 letri3 8037 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑥 = 𝑤 ↔ (𝑥𝑤𝑤𝑥)))
119, 10syl 14 . . . . . . . . 9 (((𝑥𝑆𝑤𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 = 𝑤 ↔ (𝑥𝑤𝑤𝑥)))
1211exbiri 382 . . . . . . . 8 ((𝑥𝑆𝑤𝑆) → (𝑆 ⊆ ℝ → ((𝑥𝑤𝑤𝑥) → 𝑥 = 𝑤)))
1312com23 78 . . . . . . 7 ((𝑥𝑆𝑤𝑆) → ((𝑥𝑤𝑤𝑥) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
145, 13syld 45 . . . . . 6 ((𝑥𝑆𝑤𝑆) → ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
1514com3r 79 . . . . 5 (𝑆 ⊆ ℝ → ((𝑥𝑆𝑤𝑆) → ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → 𝑥 = 𝑤)))
1615ralrimivv 2558 . . . 4 (𝑆 ⊆ ℝ → ∀𝑥𝑆𝑤𝑆 ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → 𝑥 = 𝑤))
1716anim2i 342 . . 3 ((∃𝑥𝑆𝑦𝑆 𝑥𝑦𝑆 ⊆ ℝ) → (∃𝑥𝑆𝑦𝑆 𝑥𝑦 ∧ ∀𝑥𝑆𝑤𝑆 ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → 𝑥 = 𝑤)))
1817ancoms 268 . 2 ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → (∃𝑥𝑆𝑦𝑆 𝑥𝑦 ∧ ∀𝑥𝑆𝑤𝑆 ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → 𝑥 = 𝑤)))
19 breq1 4006 . . . 4 (𝑥 = 𝑤 → (𝑥𝑦𝑤𝑦))
2019ralbidv 2477 . . 3 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑥𝑦 ↔ ∀𝑦𝑆 𝑤𝑦))
2120reu4 2931 . 2 (∃!𝑥𝑆𝑦𝑆 𝑥𝑦 ↔ (∃𝑥𝑆𝑦𝑆 𝑥𝑦 ∧ ∀𝑥𝑆𝑤𝑆 ((∀𝑦𝑆 𝑥𝑦 ∧ ∀𝑦𝑆 𝑤𝑦) → 𝑥 = 𝑤)))
2218, 21sylibr 134 1 ((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → ∃!𝑥𝑆𝑦𝑆 𝑥𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2148  wral 2455  wrex 2456  ∃!wreu 2457  wss 3129   class class class wbr 4003  cr 7809  cle 7992
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-apti 7925
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997
This theorem is referenced by:  lbcl  8902  lble  8903
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