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Theorem tz6.26i 5854
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 5853 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
41, 2, 3mpanl12 682 1 ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wne 2943  wrex 3062  wss 3723  c0 4063   Se wse 5207   We wwe 5208  Predcpred 5821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822
This theorem is referenced by:  wfrlem16  7587
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