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Theorem tz6.26 5518
Description: All nonempty (possibly proper) subclasses of 𝐴, which has a well-founded relation 𝑅, have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
tz6.26 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem tz6.26
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 wereu2 4929 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
2 reurex 3041 . . 3 (∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
31, 2syl 17 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
4 rabeq0 3814 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝑅𝑦)
5 dfrab3 3764 . . . . . 6 {𝑥𝐵𝑥𝑅𝑦} = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
6 vex 3080 . . . . . . 7 𝑦 ∈ V
76dfpred2 5496 . . . . . 6 Pred(𝑅, 𝐵, 𝑦) = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
85, 7eqtr4i 2539 . . . . 5 {𝑥𝐵𝑥𝑅𝑦} = Pred(𝑅, 𝐵, 𝑦)
98eqeq1i 2519 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
104, 9bitr3i 264 . . 3 (∀𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
1110rexbii 2927 . 2 (∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
123, 11sylib 206 1 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  {cab 2500  wne 2684  wral 2800  wrex 2801  ∃!wreu 2802  {crab 2804  cin 3443  wss 3444  c0 3777   class class class wbr 4481   Se wse 4889   We wwe 4890  Predcpred 5486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-cnv 4940  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487
This theorem is referenced by:  tz6.26i  5519  wfi  5520  wzel  30853  wsuclem  30854
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