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Theorem wereu 5100
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 5094 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 fri 5066 . . . . . 6 (((𝐵𝑉𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
32exp32 630 . . . . 5 ((𝐵𝑉𝑅 Fr 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
43expcom 451 . . . 4 (𝑅 Fr 𝐴 → (𝐵𝑉 → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))))
543imp2 1280 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
61, 5sylan 488 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
7 weso 5095 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
8 soss 5043 . . . . 5 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
97, 8mpan9 486 . . . 4 ((𝑅 We 𝐴𝐵𝐴) → 𝑅 Or 𝐵)
10 somo 5059 . . . 4 (𝑅 Or 𝐵 → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
119, 10syl 17 . . 3 ((𝑅 We 𝐴𝐵𝐴) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
12113ad2antr2 1225 . 2 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
13 reu5 3154 . 2 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
146, 12, 13sylanbrc 697 1 ((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036  wcel 1988  wne 2791  wral 2909  wrex 2910  ∃!wreu 2911  ∃*wrmo 2912  wss 3567  c0 3907   class class class wbr 4644   Or wor 5024   Fr wfr 5060   We wwe 5062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-po 5025  df-so 5026  df-fr 5063  df-we 5065
This theorem is referenced by:  htalem  8744  zorn2lem1  9303  dyadmax  23347  wessf1ornlem  39187
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