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Theorem tz6.26 6201
Description: All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. 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 5548 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
2 reurex 3338 . . 3 (∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
31, 2syl 17 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
4 rabeq0 4299 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝑅𝑦)
5 dfrab3 4224 . . . . . 6 {𝑥𝐵𝑥𝑅𝑦} = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
6 vex 3412 . . . . . . 7 𝑦 ∈ V
76dfpred2 6169 . . . . . 6 Pred(𝑅, 𝐵, 𝑦) = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
85, 7eqtr4i 2768 . . . . 5 {𝑥𝐵𝑥𝑅𝑦} = Pred(𝑅, 𝐵, 𝑦)
98eqeq1i 2742 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
104, 9bitr3i 280 . . 3 (∀𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
1110rexbii 3170 . 2 (∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
123, 11sylib 221 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 399   = wceq 1543  {cab 2714  wne 2940  wral 3061  wrex 3062  ∃!wreu 3063  {crab 3065  cin 3865  wss 3866  c0 4237   class class class wbr 5053   Se wse 5507   We wwe 5508  Predcpred 6159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160
This theorem is referenced by:  tz6.26i  6202  wfi  6203  wzel  33555  wsuclem  33556
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