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| Mirrors > Home > MPE Home > Th. List > newbday | Structured version Visualization version GIF version | ||
| Description: A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| newbday | ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | madebday 27938 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴)) | |
| 2 | oldbday 27939 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴)) | |
| 3 | 2 | notbid 318 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday ‘𝑋) ∈ 𝐴)) | 
| 4 | 1, 3 | anbi12d 632 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ((𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) | 
| 5 | newval 27894 | . . . . . 6 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) | |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) | 
| 7 | 6 | eleq2d 2827 | . . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)))) | 
| 8 | eldif 3961 | . . . 4 ⊢ (𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))) | |
| 9 | 7, 8 | bitrdi 287 | . . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))) | 
| 10 | 9 | adantr 480 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))) | 
| 11 | bdayelon 27821 | . . . . 5 ⊢ ( bday ‘𝑋) ∈ On | |
| 12 | 11 | onordi 6495 | . . . 4 ⊢ Ord ( bday ‘𝑋) | 
| 13 | eloni 6394 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 14 | ordtri4 6421 | . . . 4 ⊢ ((Ord ( bday ‘𝑋) ∧ Ord 𝐴) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) | |
| 15 | 12, 13, 14 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ On → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) | 
| 16 | 15 | adantr 480 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) | 
| 17 | 4, 10, 16 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 Ord word 6383 Oncon0 6384 ‘cfv 6561 No csur 27684 bday cbday 27686 M cmade 27881 O cold 27882 N cnew 27883 | 
| 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 df-new 27888 df-left 27889 df-right 27890 | 
| This theorem is referenced by: sltonold 28283 | 
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