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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 27741 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴)) | |
2 | oldbday 27742 | . . . 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 630 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ((𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) |
5 | newval 27697 | . . . . . 6 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
7 | 6 | eleq2d 2818 | . . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)))) |
8 | eldif 3958 | . . . 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 27624 | . . . . 5 ⊢ ( bday ‘𝑋) ∈ On | |
12 | 11 | onordi 6475 | . . . 4 ⊢ Ord ( bday ‘𝑋) |
13 | eloni 6374 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
14 | ordtri4 6401 | . . . 4 ⊢ ((Ord ( bday ‘𝑋) ∧ Ord 𝐴) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) | |
15 | 12, 13, 14 | sylancr 586 | . . 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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ⊆ wss 3948 Ord word 6363 Oncon0 6364 ‘cfv 6543 No csur 27488 bday cbday 27490 M cmade 27684 O cold 27685 N cnew 27686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-1o 8472 df-2o 8473 df-no 27491 df-slt 27492 df-bday 27493 df-sslt 27629 df-scut 27631 df-made 27689 df-old 27690 df-new 27691 df-left 27692 df-right 27693 |
This theorem is referenced by: sltonold 28068 |
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