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Theorem tz6.26OLD 6350
Description: Obsolete proof of tz6.26 6349 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
tz6.26OLD (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem tz6.26OLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 wereu2 5674 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
2 reurex 3381 . . 3 (∃!𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
31, 2syl 17 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦)
4 rabeq0 4385 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝑅𝑦)
5 dfrab3 4310 . . . . . 6 {𝑥𝐵𝑥𝑅𝑦} = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
6 vex 3479 . . . . . . 7 𝑦 ∈ V
76dfpred2 6311 . . . . . 6 Pred(𝑅, 𝐵, 𝑦) = (𝐵 ∩ {𝑥𝑥𝑅𝑦})
85, 7eqtr4i 2764 . . . . 5 {𝑥𝐵𝑥𝑅𝑦} = Pred(𝑅, 𝐵, 𝑦)
98eqeq1i 2738 . . . 4 ({𝑥𝐵𝑥𝑅𝑦} = ∅ ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
104, 9bitr3i 277 . . 3 (∀𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
1110rexbii 3095 . 2 (∃𝑦𝐵𝑥𝐵 ¬ 𝑥𝑅𝑦 ↔ ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
123, 11sylib 217 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 397   = wceq 1542  {cab 2710  wne 2941  wral 3062  wrex 3071  ∃!wreu 3375  {crab 3433  cin 3948  wss 3949  c0 4323   class class class wbr 5149   Se wse 5630   We wwe 5631  Predcpred 6300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301
This theorem is referenced by: (None)
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