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Theorem wereu 5139
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 5133 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 fri 5105 . . . . . 6 (((𝐵𝑉𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
32exp32 630 . . . . 5 ((𝐵𝑉𝑅 Fr 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
43expcom 450 . . . 4 (𝑅 Fr 𝐴 → (𝐵𝑉 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))))
543imp2 1304 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
61, 5sylan 487 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
7 weso 5134 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
8 soss 5082 . . . . 5 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
97, 8mpan9 485 . . . 4 ((𝑅 We 𝐴𝐵𝐴) → 𝑅 Or 𝐵)
10 somo 5098 . . . 4 (𝑅 Or 𝐵 → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
119, 10syl 17 . . 3 ((𝑅 We 𝐴𝐵𝐴) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
12113ad2antr2 1247 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
13 reu5 3189 . 2 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
146, 12, 13sylanbrc 699 1 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944  wss 3607  c0 3948   class class class wbr 4685   Or wor 5063   Fr wfr 5099   We wwe 5101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-po 5064  df-so 5065  df-fr 5102  df-we 5104
This theorem is referenced by:  htalem  8797  zorn2lem1  9356  dyadmax  23412  wessf1ornlem  39685
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