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Theorem tz6.26 6154
 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 5520 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
2 reurex 3377 . . 3 (∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
31, 2syl 17 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
4 rabeq0 4295 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝑅𝑦)
5 dfrab3 4233 . . . . . 6 {𝑥𝐵𝑥𝑅𝑦} = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
6 vex 3445 . . . . . . 7 𝑦 ∈ V
76dfpred2 6132 . . . . . 6 Pred(𝑅, 𝐵, 𝑦) = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
85, 7eqtr4i 2824 . . . . 5 {𝑥𝐵𝑥𝑅𝑦} = Pred(𝑅, 𝐵, 𝑦)
98eqeq1i 2803 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
104, 9bitr3i 280 . . 3 (∀𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
1110rexbii 3211 . 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 1538  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  {crab 3110   ∩ cin 3882   ⊆ wss 3883  ∅c0 4246   class class class wbr 5034   Se wse 5480   We wwe 5481  Predcpred 6122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123 This theorem is referenced by:  tz6.26i  6155  wfi  6156  wzel  33294  wsuclem  33295
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