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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 6943 | . . 3 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ dom O ) | |
| 2 | oldf 27896 | . . . 4 ⊢ O :On⟶𝒫 No | |
| 3 | 2 | fdmi 6747 | . . 3 ⊢ dom O = On | 
| 4 | 1, 3 | eleqtrdi 2851 | . 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → 𝐴 ∈ On) | 
| 5 | elold 27908 | . . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))) | |
| 6 | madebdayim 27926 | . . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ⊆ 𝑏) | |
| 7 | 6 | ad2antll 729 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ⊆ 𝑏) | 
| 8 | simprl 771 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → 𝑏 ∈ 𝐴) | |
| 9 | bdayelon 27821 | . . . . . . 7 ⊢ ( bday ‘𝑋) ∈ On | |
| 10 | ontr2 6431 | . . . . . . 7 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝐴 ∈ On) → ((( bday ‘𝑋) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴) → ( bday ‘𝑋) ∈ 𝐴)) | |
| 11 | 9, 10 | mpan 690 | . . . . . 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 699 | . . . 4 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘𝑏))) → ( bday ‘𝑋) ∈ 𝐴) | 
| 14 | 13 | rexlimdvaa 3156 | . . 3 ⊢ (𝐴 ∈ On → (∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏) → ( bday ‘𝑋) ∈ 𝐴)) | 
| 15 | 5, 14 | sylbid 240 | . 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 395 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 𝒫 cpw 4600 dom cdm 5685 Oncon0 6384 ‘cfv 6561 No csur 27684 bday cbday 27686 M cmade 27881 O cold 27882 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 | 
| This theorem is referenced by: oldirr 27928 oldbday 27939 addsbdaylem 28049 negsproplem2 28061 negsbdaylem 28088 mulsproplem2 28143 mulsproplem3 28144 mulsproplem4 28145 mulsproplem5 28146 mulsproplem6 28147 mulsproplem7 28148 mulsproplem8 28149 | 
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