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Theorem frminex 5054
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 3932 . 2 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
2 frminex.1 . . . . 5 𝐴 ∈ V
32rabex 4773 . . . 4 {𝑥𝐴𝜑} ∈ V
4 ssrab2 3666 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
5 fri 5036 . . . . . 6 ((({𝑥𝐴𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧)
6 frminex.2 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜓))
76ralrab 3350 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
87rexbii 3034 . . . . . . 7 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
9 breq2 4617 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑦𝑅𝑧𝑦𝑅𝑥))
109notbid 308 . . . . . . . . . 10 (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
1110imbi2d 330 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥)))
1211ralbidv 2980 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1312rexrab2 3356 . . . . . . 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 868 . . . 4 ((({𝑥𝐴𝜑} ∈ V ∧ {𝑥𝐴𝜑} ⊆ 𝐴) ∧ (𝑅 Fr 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
173, 4, 16mpanl12 717 . . 3 ((𝑅 Fr 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1817ex 450 . 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 384  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  c0 3891   class class class wbr 4613   Fr wfr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-fr 5033
This theorem is referenced by: (None)
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