Theorem oldbdayim 28155

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.

Step	Hyp	Ref	Expression
1		elfvdm 6916	. . 3 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ dom O )
2		oldf 28103	. . . 4 ⊢ O :On⟶𝒫 No
3	2	fdmi 6718	. . 3 ⊢ dom O = On
4	1, 3	eleqtrdi 2872	. 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ On)
5		elold 28125	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
6		madebdayim 28154	. . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ⊆ 𝑏)
7	6	ad2antll 742	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ⊆ 𝑏)
8		simprl 783	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → 𝑏 ∈ 𝐴)
9		bdayon 28018	. . . . . . 7 ⊢ ( bday ‘𝑋) ∈ On
10		ontr2 6410	. . . . . . 7 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴))
11	9, 10	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ On → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴))
12	11	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴))
13	7, 8, 12	mp2and 712	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ∈ 𝐴)
14	13	rexlimdvaa 3166	. . 3 ⊢ (𝐴 ∈ On → (∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ∈ 𝐴))
15	5, 14	sylbid 243	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴))
16	4, 15	mpcom 39	1 ⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902  𝒫 cpw 4560  dom cdm 5659  Oncon0 6361  ‘cfv 6537   No csur 27877   bday cbday 27879   M cmade 28088   O cold 28089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8458  df-2o 8459  df-no 27880  df-lts 27881  df-bday 27882  df-slts 28024  df-cuts 28026  df-made 28093  df-old 28094

This theorem is used by:  oldirr  28156  oldbday  28167  bdayiun  28181  addbdaylem  28283  negsproplem2  28295  negbdaylem  28322  mulsproplem2  28383  mulsproplem3  28384  mulsproplem4  28385  mulsproplem5  28386  mulsproplem6  28387  mulsproplem7  28388  mulsproplem8  28389  oniso  28537  bdayfinbndlem1  28733