Theorem newbday 27894

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 27892	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
2		oldbday 27893	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))
3	2	notbid 318	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday ‘𝑋) ∈ 𝐴))
4	1, 3	anbi12d 633	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ((𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)) ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴)))
5		newval 27827	. . . . . 6 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
7	6	eleq2d 2822	. . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴))))
8		eldif 3899	. . . 4 ⊢ (𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))
9	7, 8	bitrdi 287	. . 3 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
10	9	adantr 480	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
11		bdayon 27744	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
12	11	onordi 6436	. . . 4 ⊢ Ord ( bday ‘𝑋)
13		eloni 6333	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
14		ordtri4 6360	. . . 4 ⊢ ((Ord ( bday ‘𝑋) ∧ Ord 𝐴) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴)))
15	12, 13, 14	sylancr 588	. . 3 ⊢ (𝐴 ∈ On → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴)))
16	15	adantr 480	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) = 𝐴 ↔ (( bday ‘𝑋) ⊆ 𝐴 ∧ ¬ ( bday ‘𝑋) ∈ 𝐴)))
17	4, 10, 16	3bitr4d 311	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886   ⊆ wss 3889  Ord word 6322  Oncon0 6323  ‘cfv 6498   No csur 27603   bday cbday 27605   M cmade 27814   O cold 27815   N cnew 27816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-new 27821  df-left 27822  df-right 27823

This theorem is referenced by:  newbdayim  27895  ltonold  28253