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

Theorem wereu 5534
Description: A subset of a well-ordered set has a unique 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
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem wereu
StepHypRef Expression
1 wefr 5528 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 fri 5500 . . . . . 6 (((𝐵𝑉𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
32exp32 424 . . . . 5 ((𝐵𝑉𝑅 Fr 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
43expcom 417 . . . 4 (𝑅 Fr 𝐴 → (𝐵𝑉 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))))
543imp2 1346 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
61, 5sylan 583 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
7 weso 5529 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
8 soss 5476 . . . . 5 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
97, 8mpan9 510 . . . 4 ((𝑅 We 𝐴𝐵𝐴) → 𝑅 Or 𝐵)
10 somo 5493 . . . 4 (𝑅 Or 𝐵 → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
119, 10syl 17 . . 3 ((𝑅 We 𝐴𝐵𝐴) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
12113ad2antr2 1186 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
13 reu5 3413 . 2 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
146, 12, 13sylanbrc 586 1 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084  wcel 2115  wne 3013  wral 3132  wrex 3133  ∃!wreu 3134  ∃*wrmo 3135  wss 3918  c0 4274   class class class wbr 5049   Or wor 5456   Fr wfr 5494   We wwe 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5050  df-po 5457  df-so 5458  df-fr 5497  df-we 5499
This theorem is referenced by:  htalem  9311  zorn2lem1  9905  dyadmax  24193  finorwe  34706  wessf1ornlem  41667
  Copyright terms: Public domain W3C validator