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Theorem newbday 27955
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 27953 . . 3 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday 𝑋) ⊆ 𝐴))
2 oldbday 27954 . . . 4 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday 𝑋) ∈ 𝐴))
32notbid 318 . . 3 ((𝐴 ∈ On ∧ 𝑋 No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday 𝑋) ∈ 𝐴))
41, 3anbi12d 632 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → ((𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)) ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
5 newval 27909 . . . . . 6 ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
65a1i 11 . . . . 5 (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
76eleq2d 2825 . . . 4 (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴))))
8 eldif 3973 . . . 4 (𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))
97, 8bitrdi 287 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
109adantr 480 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
11 bdayelon 27836 . . . . 5 ( bday 𝑋) ∈ On
1211onordi 6497 . . . 4 Ord ( bday 𝑋)
13 eloni 6396 . . . 4 (𝐴 ∈ On → Ord 𝐴)
14 ordtri4 6423 . . . 4 ((Ord ( bday 𝑋) ∧ Ord 𝐴) → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
1512, 13, 14sylancr 587 . . 3 (𝐴 ∈ On → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
1615adantr 480 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
174, 10, 163bitr4d 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 1537  wcel 2106  cdif 3960  wss 3963  Ord word 6385  Oncon0 6386  cfv 6563   No csur 27699   bday cbday 27701   M cmade 27896   O cold 27897   N cnew 27898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-new 27903  df-left 27904  df-right 27905
This theorem is referenced by:  sltonold  28298
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