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Theorem newbday 27743
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 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))
32notbid 318 . . 3 ((𝐴 ∈ On ∧ 𝑋 No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday 𝑋) ∈ 𝐴))
41, 3anbi12d 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))
65a1i 11 . . . . 5 (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
76eleq2d 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)))
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 27624 . . . . 5 ( bday 𝑋) ∈ On
1211onordi 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)))
1512, 13, 14sylancr 586 . . 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 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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