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Theorem madebdaylemold 33734
Description: Lemma for madebday 33736. If the inductive hypothesis of madebday 33736 is satisfied, the converse of oldbdayim 33727 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
Assertion
Ref Expression
madebdaylemold ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( bday 𝑋) ∈ 𝐴𝑋 ∈ ( O ‘𝐴)))
Distinct variable groups:   𝐴,𝑏   𝑦,𝑏,𝑋
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem madebdaylemold
StepHypRef Expression
1 fveq2 6687 . . . . . . . . 9 (𝑦 = 𝑋 → ( bday 𝑦) = ( bday 𝑋))
21sseq1d 3918 . . . . . . . 8 (𝑦 = 𝑋 → (( bday 𝑦) ⊆ 𝑏 ↔ ( bday 𝑋) ⊆ 𝑏))
3 eleq1 2821 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑋 ∈ ( M ‘𝑏)))
42, 3imbi12d 348 . . . . . . 7 (𝑦 = 𝑋 → ((( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ↔ (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏))))
54rspcv 3524 . . . . . 6 (𝑋 No → (∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) → (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏))))
65ralimdv 3093 . . . . 5 (𝑋 No → (∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) → ∀𝑏𝐴 (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏))))
76impcom 411 . . . 4 ((∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ∀𝑏𝐴 (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏)))
8 rexim 3155 . . . 4 (∀𝑏𝐴 (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏)) → (∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏 → ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
97, 8syl 17 . . 3 ((∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏 → ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
1093adant1 1131 . 2 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏 → ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
11 bdayelon 33627 . . . 4 ( bday 𝑋) ∈ On
12 onelssex 33247 . . . 4 ((( bday 𝑋) ∈ On ∧ 𝐴 ∈ On) → (( bday 𝑋) ∈ 𝐴 ↔ ∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏))
1311, 12mpan 690 . . 3 (𝐴 ∈ On → (( bday 𝑋) ∈ 𝐴 ↔ ∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏))
14133ad2ant1 1134 . 2 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( bday 𝑋) ∈ 𝐴 ↔ ∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏))
15 elold 33709 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
16153ad2ant1 1134 . 2 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
1710, 14, 163imtr4d 297 1 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( bday 𝑋) ∈ 𝐴𝑋 ∈ ( O ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  wss 3853  Oncon0 6183  cfv 6350   No csur 33499   bday cbday 33501   M cmade 33682   O cold 33683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6186  df-on 6187  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7140  df-ov 7186  df-oprab 7187  df-mpo 7188  df-wrecs 7989  df-recs 8050  df-1o 8144  df-2o 8145  df-no 33502  df-slt 33503  df-bday 33504  df-sslt 33632  df-scut 33634  df-made 33687  df-old 33688
This theorem is referenced by:  madebdaylemlrcut  33735  oldbday  33737
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