Theorem oldbday 27982

Description: A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)

Assertion

Ref	Expression
oldbday	⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))

Proof of Theorem oldbday

Dummy variables 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oldbdayim 27970	. 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴)
2		simpl 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → 𝐴 ∈ On)
3		onelon 6366	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ On)
4		madebday 27981	. . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( M ‘𝑏) ↔ ( bday ‘𝑦) ⊆ 𝑏))
5	4	biimprd 250	. . . . . . 7 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
6	3, 5	sylan 589	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
7	6	anasss 470	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑦 ∈ No )) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
8	7	ralrimivva 3204	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
9	8	adantr 484	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
10		simpr 488	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → 𝑋 ∈ No )
11		madebdaylemold 27979	. . 3 ⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
12	2, 9, 10, 11	syl3anc 1389	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
13	1, 12	impbid2 228	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3902  Oncon0 6341  ‘cfv 6516   No csur 27692   bday cbday 27694   M cmade 27903   O cold 27904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-1o 8431  df-2o 8432  df-no 27695  df-lts 27696  df-bday 27697  df-slts 27839  df-cuts 27841  df-made 27908  df-old 27909  df-left 27911  df-right 27912

This theorem is referenced by:  newbday  27983  0elold  27991  cofcutr  28005  lrrecval2  28021  addsproplem2  28051  addsproplem4  28053  addsproplem5  28054  addsproplem6  28055  negsproplem4  28112  negsproplem5  28113  negsproplem6  28114  negleft  28139  negright  28140  mulsproplem12  28208  mulsproplem13  28209  mulsproplem14  28210  ltonold  28342  oncutlt  28345  onnolt  28347  onlts  28348  oniso  28352  n0ssoldg  28434  onsfi  28437  bdayfinbndlem1  28548  dfz12s2  28569