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Theorem newbday 33712
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.)
Assertion
Ref Expression
newbday ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday 𝑋) = 𝐴))

Proof of Theorem newbday
StepHypRef 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))
32notbid 321 . . 3 ((𝐴 ∈ On ∧ 𝑋 No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday 𝑋) ∈ 𝐴))
41, 3anbi12d 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)))
65eleq2d 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)))
86, 7bitrdi 290 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
98adantr 484 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
10 bdayelon 33604 . . . . 5 ( bday 𝑋) ∈ On
1110onordi 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)))
1411, 12, 13sylancr 590 . . 3 (𝐴 ∈ On → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
1514adantr 484 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
164, 9, 153bitr4d 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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