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Theorem frminex 5668
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 4395 . 2 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
2 frminex.1 . . . . 5 𝐴 ∈ V
32rabex 5345 . . . 4 {𝑥𝐴𝜑} ∈ V
4 ssrab2 4090 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
5 fri 5646 . . . . . 6 ((({𝑥𝐴𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧)
6 frminex.2 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜓))
76ralrab 3702 . . . . . . . 8 (∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
87rexbii 3092 . . . . . . 7 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧))
9 breq2 5152 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑦𝑅𝑧𝑦𝑅𝑥))
109notbid 318 . . . . . . . . . 10 (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥))
1110imbi2d 340 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥)))
1211ralbidv 3176 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1312rexrab2 3709 . . . . . . 7 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
148, 13bitri 275 . . . . . 6 (∃𝑧 ∈ {𝑥𝐴𝜑}∀𝑦 ∈ {𝑥𝐴𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
155, 14sylib 218 . . . . 5 ((({𝑥𝐴𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1615an4s 660 . . . 4 ((({𝑥𝐴𝜑} ∈ V ∧ {𝑥𝐴𝜑} ⊆ 𝐴) ∧ (𝑅 Fr 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅)) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
173, 4, 16mpanl12 702 . . 3 ((𝑅 Fr 𝐴 ∧ {𝑥𝐴𝜑} ≠ ∅) → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))
1817ex 412 . 2 (𝑅 Fr 𝐴 → ({𝑥𝐴𝜑} ≠ ∅ → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
191, 18biimtrrid 243 1 (𝑅 Fr 𝐴 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148   Fr wfr 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-fr 5641
This theorem is referenced by: (None)
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