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Theorem madebdaylemold 27951
Description: Lemma for madebday 27953. If the inductive hypothesis of madebday 27953 is satisfied, the converse of oldbdayim 27942 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 6907 . . . . . . . . 9 (𝑦 = 𝑋 → ( bday 𝑦) = ( bday 𝑋))
21sseq1d 4027 . . . . . . . 8 (𝑦 = 𝑋 → (( bday 𝑦) ⊆ 𝑏 ↔ ( bday 𝑋) ⊆ 𝑏))
3 eleq1 2827 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑋 ∈ ( M ‘𝑏)))
42, 3imbi12d 344 . . . . . . 7 (𝑦 = 𝑋 → ((( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ↔ (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏))))
54rspcv 3618 . . . . . 6 (𝑋 No → (∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) → (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏))))
65ralimdv 3167 . . . . 5 (𝑋 No → (∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) → ∀𝑏𝐴 (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏))))
76impcom 407 . . . 4 ((∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ∀𝑏𝐴 (( bday 𝑋) ⊆ 𝑏𝑋 ∈ ( M ‘𝑏)))
8 rexim 3085 . . . 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 1129 . 2 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏 → ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
11 bdayelon 27836 . . . 4 ( bday 𝑋) ∈ On
12 onelssex 6434 . . . 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 1132 . 2 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( bday 𝑋) ∈ 𝐴 ↔ ∃𝑏𝐴 ( bday 𝑋) ⊆ 𝑏))
15 elold 27923 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
16153ad2ant1 1132 . 2 ((𝐴 ∈ On ∧ ∀𝑏𝐴𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
1710, 14, 163imtr4d 294 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  Oncon0 6386  cfv 6563   No csur 27699   bday cbday 27701   M cmade 27896   O cold 27897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902
This theorem is referenced by:  madebdaylemlrcut  27952  oldbday  27954
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