Theorem newbday 27911

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 27909	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
2		oldbday 27910	. . . 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 27844	. . . . . 6 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
7	6	eleq2d 2823	. . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴))))
8		eldif 3900	. . . 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 27761	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
12	11	onordi 6431	. . . 4 ⊢ Ord ( bday ‘𝑋)
13		eloni 6328	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
14		ordtri4 6355	. . . 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 3887   ⊆ wss 3890  Ord word 6317  Oncon0 6318  ‘cfv 6493   No csur 27620   bday cbday 27622   M cmade 27831   O cold 27832   N cnew 27833

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27623  df-lts 27624  df-bday 27625  df-slts 27767  df-cuts 27769  df-made 27836  df-old 27837  df-new 27838  df-left 27839  df-right 27840

This theorem is referenced by:  newbdayim  27912  ltonold  28270