MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wereu Structured version   Visualization version   GIF version

Theorem wereu 5685
Description: A nonempty subset of an 𝑅-well-ordered class has a unique 𝑅 -minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
wereu ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑉,𝑦

Proof of Theorem wereu
StepHypRef Expression
1 wefr 5679 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 fri 5646 . . . . . 6 (((𝐵𝑉𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
32exp32 420 . . . . 5 ((𝐵𝑉𝑅 Fr 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
43expcom 413 . . . 4 (𝑅 Fr 𝐴 → (𝐵𝑉 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))))
543imp2 1348 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
61, 5sylan 580 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
7 weso 5680 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
8 soss 5617 . . . . 5 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
97, 8mpan9 506 . . . 4 ((𝑅 We 𝐴𝐵𝐴) → 𝑅 Or 𝐵)
10 somo 5635 . . . 4 (𝑅 Or 𝐵 → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
119, 10syl 17 . . 3 ((𝑅 We 𝐴𝐵𝐴) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
12113ad2antr2 1188 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
13 reu5 3380 . 2 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
146, 12, 13sylanbrc 583 1 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086  wcel 2106  wne 2938  wral 3059  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  wss 3963  c0 4339   class class class wbr 5148   Or wor 5596   Fr wfr 5638   We wwe 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-po 5597  df-so 5598  df-fr 5641  df-we 5643
This theorem is referenced by:  htalem  9934  zorn2lem1  10534  dyadmax  25647  wevgblacfn  35093  finorwe  37365  wessf1ornlem  45128
  Copyright terms: Public domain W3C validator