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Mirrors > Home > MPE Home > Th. List > oldbdayim | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
oldbdayim | ⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6876 | . . 3 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ dom O ) | |
2 | oldf 27138 | . . . 4 ⊢ O :On⟶𝒫 No | |
3 | 2 | fdmi 6677 | . . 3 ⊢ dom O = On |
4 | 1, 3 | eleqtrdi 2848 | . 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ On) |
5 | elold 27150 | . . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) | |
6 | madebdayim 27167 | . . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ⊆ 𝑏) | |
7 | 6 | ad2antll 727 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ⊆ 𝑏) |
8 | simprl 769 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → 𝑏 ∈ 𝐴) | |
9 | bdayelon 27067 | . . . . . . 7 ⊢ ( bday ‘𝑋) ∈ On | |
10 | ontr2 6362 | . . . . . . 7 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴)) | |
11 | 9, 10 | mpan 688 | . . . . . 6 ⊢ (𝐴 ∈ On → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴)) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴)) |
13 | 7, 8, 12 | mp2and 697 | . . . 4 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ∈ 𝐴) |
14 | 13 | rexlimdvaa 3151 | . . 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) |
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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