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Theorem tz6.26i 5629
 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, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
tz6.26i.1 𝑅 We 𝐴
tz6.26i.2 𝑅 Se 𝐴
Assertion
Ref Expression
tz6.26i ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem tz6.26i
StepHypRef Expression
1 tz6.26i.1 . 2 𝑅 We 𝐴
2 tz6.26i.2 . 2 𝑅 Se 𝐴
3 tz6.26 5628 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
41, 2, 3mpanl12 714 1 ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   Se wse 4995   We wwe 4996  Predcpred 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597 This theorem is referenced by:  wfrlem16  7317
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