Theorem oldbdayim 33998

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 6788	. . 3 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ dom O )
2		oldf 33968	. . . 4 ⊢ O :On⟶𝒫 No
3	2	fdmi 6596	. . 3 ⊢ dom O = On
4	1, 3	eleqtrdi 2849	. 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ On)
5		elold 33980	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
6		madebdayim 33997	. . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ⊆ 𝑏)
7	6	ad2antll 725	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ⊆ 𝑏)
8		simprl 767	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → 𝑏 ∈ 𝐴)
9		bdayelon 33898	. . . . . . 7 ⊢ ( bday ‘𝑋) ∈ On
10		ontr2 6298	. . . . . . 7 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴))
11	9, 10	mpan 686	. . . . . 6 ⊢ (𝐴 ∈ On → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴))
12	11	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴))
13	7, 8, 12	mp2and 695	. . . 4 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ∈ 𝐴)
14	13	rexlimdvaa 3213	. . 3 ⊢ (𝐴 ∈ On → (∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ∈ 𝐴))
15	5, 14	sylbid 239	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴))
16	4, 15	mpcom 38	1 ⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883  𝒫 cpw 4530  dom cdm 5580  Oncon0 6251  ‘cfv 6418   No csur 33770   bday cbday 33772   M cmade 33953   O cold 33954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959

This theorem is referenced by:  oldirr  33999  oldbday  34008