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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33710 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴)) | |
2 | oldbday 33711 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴)) | |
3 | 2 | notbid 321 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday ‘𝑋) ∈ 𝐴)) |
4 | 1, 3 | anbi12d 634 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ((𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) |
5 | newval 33672 | . . . . 5 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) | |
6 | 5 | eleq2d 2818 | . . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)))) |
7 | eldif 3851 | . . . 4 ⊢ (𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))) | |
8 | 6, 7 | bitrdi 290 | . . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))) |
9 | 8 | adantr 484 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))) |
10 | bdayelon 33604 | . . . . 5 ⊢ ( bday ‘𝑋) ∈ On | |
11 | 10 | onordi 6271 | . . . 4 ⊢ Ord ( bday ‘𝑋) |
12 | eloni 6176 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
13 | ordtri4 6203 | . . . 4 ⊢ ((Ord ( bday ‘𝑋) ∧ Ord 𝐴) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) | |
14 | 11, 12, 13 | sylancr 590 | . . 3 ⊢ (𝐴 ∈ On → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) |
15 | 14 | adantr 484 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴))) |
16 | 4, 9, 15 | 3bitr4d 314 | 1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∖ cdif 3838 ⊆ wss 3841 Ord word 6165 Oncon0 6166 ‘cfv 6333 No csur 33476 bday cbday 33478 M cmade 33659 O cold 33660 N cnew 33661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-wrecs 7969 df-recs 8030 df-1o 8124 df-2o 8125 df-no 33479 df-slt 33480 df-bday 33481 df-sslt 33609 df-scut 33611 df-made 33664 df-old 33665 df-new 33666 df-left 33667 df-right 33668 |
This theorem is referenced by: (None) |
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