Theorem newbday 28297

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

Step	Hyp	Ref	Expression
1		madebday 28295	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No) → (𝑋 ∈ (M‘𝐴) ↔ (bday‘𝑋) ⊆ 𝐴))
2		oldbday 28296	. . . 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 644	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No) → ((𝑋 ∈ (M‘𝐴) ∧ ¬ 𝑋 ∈ (O‘𝐴)) ↔ ((bday‘𝑋) ⊆ 𝐴 ∧ ¬ (bday‘𝑋) ∈ 𝐴)))
5		newval 28230	. . . . . 6 ⊢ (N‘𝐴) = ((M‘𝐴) ∖ (O‘𝐴))
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → (N‘𝐴) = ((M‘𝐴) ∖ (O‘𝐴)))
7	6	eleq2d 2846	. . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ (N‘𝐴) ↔ 𝑋 ∈ ((M‘𝐴) ∖ (O‘𝐴))))
8		eldif 4048	. . . 4 ⊢ (𝑋 ∈ ((M‘𝐴) ∖ (O‘𝐴)) ↔ (𝑋 ∈ (M‘𝐴) ∧ ¬ 𝑋 ∈ (O‘𝐴)))
9	7, 8	bitrdi 290	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ (N‘𝐴) ↔ (𝑋 ∈ (M‘𝐴) ∧ ¬ 𝑋 ∈ (O‘𝐴))))
10	9	adantr 486	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No) → (𝑋 ∈ (N‘𝐴) ↔ (𝑋 ∈ (M‘𝐴) ∧ ¬ 𝑋 ∈ (O‘𝐴))))
11		bdayon 28147	. . . . 5 ⊢ (bday‘𝑋) ∈ On
12	11	onordi 6479	. . . 4 ⊢ Ord (bday‘𝑋)
13		eloni 6375	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
14		ordtri4 6403	. . . 4 ⊢ ((Ord (bday‘𝑋) ∧ Ord 𝐴) → ((bday‘𝑋) = 𝐴 ↔ ((bday‘𝑋) ⊆ 𝐴 ∧ ¬ (bday‘𝑋) ∈ 𝐴)))
15	12, 13, 14	sylancr 599	. . 3 ⊢ (𝐴 ∈ On → ((bday‘𝑋) = 𝐴 ↔ ((bday‘𝑋) ⊆ 𝐴 ∧ ¬ (bday‘𝑋) ∈ 𝐴)))
16	15	adantr 486	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No) → ((bday‘𝑋) = 𝐴 ↔ ((bday‘𝑋) ⊆ 𝐴 ∧ ¬ (bday‘𝑋) ∈ 𝐴)))
17	4, 10, 16	3bitr4d 314	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No) → (𝑋 ∈ (N‘𝐴) ↔ (bday‘𝑋) = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∖ cdif 4044  Ord word 6364  Oncon0 6365  ‘cfv 6543  Nocsur 28006  bdaycbday 28008  Mcmade 28217  Ocold 28218  Ncnew 28219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223  df-new 28224  df-left 28225  df-right 28226

This theorem is used by:  newbdayim  28298  ltonold  28656