Theorem newbday 28345

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 28343	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No) → (𝑋 ∈ (M‘𝐴) ↔ (bday‘𝑋) ⊆ 𝐴))
2		oldbday 28344	. . . 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 28278	. . . . . 6 ⊢ (N‘𝐴) = ((M‘𝐴) ∖ (O‘𝐴))
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → (N‘𝐴) = ((M‘𝐴) ∖ (O‘𝐴)))
7	6	eleq2d 2850	. . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ (N‘𝐴) ↔ 𝑋 ∈ ((M‘𝐴) ∖ (O‘𝐴))))
8		eldif 4052	. . . 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 28195	. . . . 5 ⊢ (bday‘𝑋) ∈ On
12	11	onordi 6502	. . . 4 ⊢ Ord (bday‘𝑋)
13		eloni 6398	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
14		ordtri4 6426	. . . 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 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048  Ord word 6387  Oncon0 6388  ‘cfv 6566  Nocsur 28054  bdaycbday 28056  Mcmade 28265  Ocold 28266  Ncnew 28267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271  df-new 28272  df-left 28273  df-right 28274

This theorem is used by:  newbdayim  28346  ltonold  28704