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Theorem wereu 5585
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 5579 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 fri 5549 . . . . . 6 (((𝐵𝑉𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
32exp32 421 . . . . 5 ((𝐵𝑉𝑅 Fr 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
43expcom 414 . . . 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 5580 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
8 soss 5523 . . . . 5 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
97, 8mpan9 507 . . . 4 ((𝑅 We 𝐴𝐵𝐴) → 𝑅 Or 𝐵)
10 somo 5540 . . . 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 3360 . 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 396  w3a 1086  wcel 2110  wne 2945  wral 3066  wrex 3067  ∃!wreu 3068  ∃*wrmo 3069  wss 3892  c0 4262   class class class wbr 5079   Or wor 5502   Fr wfr 5541   We wwe 5543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-po 5503  df-so 5504  df-fr 5544  df-we 5546
This theorem is referenced by:  htalem  9653  zorn2lem1  10251  dyadmax  24758  finorwe  35547  wessf1ornlem  42690
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