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Theorem oldbdayim 33628
 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 ((𝐴 ∈ On ∧ 𝑋 ∈ ( O ‘𝐴)) → ( bday 𝑋) ∈ 𝐴)

Proof of Theorem oldbdayim
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 elold 33609 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
2 onelon 6194 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴) → 𝑏 ∈ On)
32adantrr 716 . . . . . 6 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → 𝑏 ∈ On)
4 simprr 772 . . . . . 6 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → 𝑋 ∈ ( M ‘𝑏))
5 madebdayim 33627 . . . . . 6 ((𝑏 ∈ On ∧ 𝑋 ∈ ( M ‘𝑏)) → ( bday 𝑋) ⊆ 𝑏)
63, 4, 5syl2anc 587 . . . . 5 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → ( bday 𝑋) ⊆ 𝑏)
7 simprl 770 . . . . 5 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → 𝑏𝐴)
8 bdayelon 33536 . . . . . . 7 ( bday 𝑋) ∈ On
9 ontr2 6216 . . . . . . 7 ((( bday 𝑋) ∈ On ∧ 𝐴 ∈ On) → ((( bday 𝑋) ⊆ 𝑏𝑏𝐴) → ( bday 𝑋) ∈ 𝐴))
108, 9mpan 689 . . . . . 6 (𝐴 ∈ On → ((( bday 𝑋) ⊆ 𝑏𝑏𝐴) → ( bday 𝑋) ∈ 𝐴))
1110adantr 484 . . . . 5 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → ((( bday 𝑋) ⊆ 𝑏𝑏𝐴) → ( bday 𝑋) ∈ 𝐴))
126, 7, 11mp2and 698 . . . 4 ((𝐴 ∈ On ∧ (𝑏𝐴𝑋 ∈ ( M ‘𝑏))) → ( bday 𝑋) ∈ 𝐴)
1312rexlimdvaa 3209 . . 3 (𝐴 ∈ On → (∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏) → ( bday 𝑋) ∈ 𝐴))
141, 13sylbid 243 . 2 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) → ( bday 𝑋) ∈ 𝐴))
1514imp 410 1 ((𝐴 ∈ On ∧ 𝑋 ∈ ( O ‘𝐴)) → ( bday 𝑋) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3858  Oncon0 6169  ‘cfv 6335   bday cbday 33410   M cmade 33586   O cold 33587 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7957  df-recs 8018  df-1o 8112  df-2o 8113  df-no 33411  df-slt 33412  df-bday 33413  df-sslt 33541  df-scut 33543  df-made 33591  df-old 33592 This theorem is referenced by:  oldirr  33629  oldbday  33638
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