Theorem oldbday 28094

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 28082	. 2 ⊢ (𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴)
2		simpl 487	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → 𝐴 ∈ On)
3		onelon 6385	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ On)
4		madebday 28093	. . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ No ) → (𝑦 ∈ ( M ‘𝑏) ↔ ( bday ‘𝑦) ⊆ 𝑏))
5	4	biimprd 251	. . . . . . 7 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
6	3, 5	sylan 591	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
7	6	anasss 471	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝑏 ∈ 𝐴 ∧ 𝑦 ∈ No )) → (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
8	7	ralrimivva 3208	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
9	8	adantr 485	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
10		simpr 489	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → 𝑋 ∈ No )
11		madebdaylemold 28091	. . 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 1398	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
13	1, 12	impbid2 229	1 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  Oncon0 6360  ‘cfv 6536   No csur 27804   bday cbday 27806   M cmade 28015   O cold 28016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-no 27807  df-lts 27808  df-bday 27809  df-slts 27951  df-cuts 27953  df-made 28020  df-old 28021  df-left 28023  df-right 28024

This theorem is referenced by:  newbday  28095  0elold  28103  cofcutr  28117  lrrecval2  28133  addsproplem2  28163  addsproplem4  28165  addsproplem5  28166  addsproplem6  28167  negsproplem4  28224  negsproplem5  28225  negsproplem6  28226  negleft  28251  negright  28252  mulsproplem12  28320  mulsproplem13  28321  mulsproplem14  28322  ltonold  28454  oncutlt  28457  onnolt  28459  onlts  28460  oniso  28464  n0ssoldg  28546  onsfi  28549  bdayfinbndlem1  28660  dfz12s2  28681