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Theorem oldbdayim 27168
Description: If 𝑋 is in the old set for 𝐴, then the birthday of 𝑋 is less than 𝐴. (Contributed by Scott Fenton, 10-Aug-2024.)
Assertion
Ref Expression
oldbdayim (𝑋 ∈ ( O ‘𝐴) → ( bday 𝑋) ∈ 𝐴)

Proof of Theorem oldbdayim
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6876 . . 3 (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ dom O )
2 oldf 27138 . . . 4 O :On⟶𝒫 No
32fdmi 6677 . . 3 dom O = On
41, 3eleqtrdi 2848 . 2 (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ On)
5 elold 27150 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
6 madebdayim 27167 . . . . . 6 (𝑋 ∈ ( M ‘𝑏) → ( bday 𝑋) ⊆ 𝑏)
76ad2antll 727 . . . . 5 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → ( bday 𝑋) ⊆ 𝑏)
8 simprl 769 . . . . 5 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → 𝑏𝐴)
9 bdayelon 27067 . . . . . . 7 ( bday 𝑋) ∈ On
10 ontr2 6362 . . . . . . 7 ((( bday 𝑋) ∈ On ∧ 𝐴 ∈ On) → ((( bday 𝑋) ⊆ 𝑏𝑏𝐴) → ( bday 𝑋) ∈ 𝐴))
119, 10mpan 688 . . . . . 6 (𝐴 ∈ On → ((( bday 𝑋) ⊆ 𝑏𝑏𝐴) → ( bday 𝑋) ∈ 𝐴))
1211adantr 481 . . . . 5 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → ((( bday 𝑋) ⊆ 𝑏𝑏𝐴) → ( bday 𝑋) ∈ 𝐴))
137, 8, 12mp2and 697 . . . 4 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → ( bday 𝑋) ∈ 𝐴)
1413rexlimdvaa 3151 . . 3 (𝐴 ∈ On → (∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏) → ( bday 𝑋) ∈ 𝐴))
155, 14sylbid 239 . 2 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) → ( bday 𝑋) ∈ 𝐴))
164, 15mpcom 38 1 (𝑋 ∈ ( O ‘𝐴) → ( bday 𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wrex 3071  wss 3908  𝒫 cpw 4558  dom cdm 5631  Oncon0 6315  cfv 6493   No csur 26939   bday cbday 26941   M cmade 27123   O cold 27124
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-1o 8404  df-2o 8405  df-no 26942  df-slt 26943  df-bday 26944  df-sslt 27072  df-scut 27074  df-made 27128  df-old 27129
This theorem is referenced by:  oldirr  27169  oldbday  27178  negsproplem2  34315
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