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Theorem frminex 5229
Description: If an element of a well-founded set satisfies a property 𝜑, then there is a minimal element that satisfies 𝜑. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
frminex.1 𝐴 ∈ V
frminex.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
frminex (𝑅 Fr 𝐴 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem frminex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rabn0 4102 . 2 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
2 frminex.1 . . . . 5 𝐴 ∈ V
32rabex 4943 . . . 4 {𝑥𝐴𝜑} ∈ V
4 ssrab2 3834 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
5 fri 5211 . . . . . 6 ((({𝑥𝐴𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧)
6 frminex.2 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜓))
76ralrab 3518 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
87rexbii 3188 . . . . . . 7 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
9 breq2 4788 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑦𝑅𝑧𝑦𝑅𝑥))
109notbid 307 . . . . . . . . . 10 (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
1110imbi2d 329 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥)))
1211ralbidv 3134 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1312rexrab2 3524 . . . . . . 7 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
148, 13bitri 264 . . . . . 6 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
155, 14sylib 208 . . . . 5 ((({𝑥𝐴𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1615an4s 631 . . . 4 ((({𝑥𝐴𝜑} ∈ V ∧ {𝑥𝐴𝜑} ⊆ 𝐴) ∧ (𝑅 Fr 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
173, 4, 16mpanl12 674 . . 3 ((𝑅 Fr 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1817ex 397 . 2 (𝑅 Fr 𝐴 → ({𝑥𝐴𝜑} ≠ ∅ → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
191, 18syl5bir 233 1 (𝑅 Fr 𝐴 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wcel 2144  wne 2942  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  wss 3721  c0 4061   class class class wbr 4784   Fr wfr 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-fr 5208
This theorem is referenced by: (None)
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