Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  scutbdaybnd2lim Structured version   Visualization version   GIF version

Theorem scutbdaybnd2lim 34011
Description: An upper bound on the birthday of a surreal cut when it is a limit birthday. (Contributed by Scott Fenton, 7-Aug-2024.)
Assertion
Ref Expression
scutbdaybnd2lim ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ (𝐴𝐵)))

Proof of Theorem scutbdaybnd2lim
StepHypRef Expression
1 scutbdaybnd2 34010 . . . 4 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21adantr 481 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
3 bdayfun 33967 . . . . . . . . 9 Fun bday
4 ssltex1 33981 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
5 ssltex2 33982 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐵 ∈ V)
6 unexg 7599 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
74, 5, 6syl2anc 584 . . . . . . . . 9 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
8 funimaexg 6520 . . . . . . . . 9 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
93, 7, 8sylancr 587 . . . . . . . 8 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
109uniexd 7595 . . . . . . 7 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
1110adantr 481 . . . . . 6 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday “ (𝐴𝐵)) ∈ V)
12 nlimsucg 7689 . . . . . 6 ( ( bday “ (𝐴𝐵)) ∈ V → ¬ Lim suc ( bday “ (𝐴𝐵)))
1311, 12syl 17 . . . . 5 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ¬ Lim suc ( bday “ (𝐴𝐵)))
14 limeq 6278 . . . . . . 7 (( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)) → (Lim ( bday ‘(𝐴 |s 𝐵)) ↔ Lim suc ( bday “ (𝐴𝐵))))
1514biimpcd 248 . . . . . 6 (Lim ( bday ‘(𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)) → Lim suc ( bday “ (𝐴𝐵))))
1615adantl 482 . . . . 5 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → (( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)) → Lim suc ( bday “ (𝐴𝐵))))
1713, 16mtod 197 . . . 4 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ¬ ( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)))
1817neqned 2950 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵)))
19 bdayelon 33971 . . . . 5 ( bday ‘(𝐴 |s 𝐵)) ∈ On
2019onordi 6371 . . . 4 Ord ( bday ‘(𝐴 |s 𝐵))
21 imassrn 5980 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
22 bdayrn 33970 . . . . . . 7 ran bday = On
2321, 22sseqtri 3957 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
24 ssorduni 7629 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
2523, 24ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
26 ordsuc 7661 . . . . 5 (Ord ( bday “ (𝐴𝐵)) ↔ Ord suc ( bday “ (𝐴𝐵)))
2725, 26mpbi 229 . . . 4 Ord suc ( bday “ (𝐴𝐵))
28 ordelssne 6293 . . . 4 ((Ord ( bday ‘(𝐴 |s 𝐵)) ∧ Ord suc ( bday “ (𝐴𝐵))) → (( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵)) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵)))))
2920, 27, 28mp2an 689 . . 3 (( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵)) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵))))
302, 18, 29sylanbrc 583 . 2 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵)))
3119a1i 11 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ∈ On)
32 ordsssuc 6352 . . 3 ((( bday ‘(𝐴 |s 𝐵)) ∈ On ∧ Ord ( bday “ (𝐴𝐵))) → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ (𝐴𝐵)) ↔ ( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵))))
3331, 25, 32sylancl 586 . 2 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ (𝐴𝐵)) ↔ ( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵))))
3430, 33mpbird 256 1 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  Vcvv 3432  cun 3885  wss 3887   cuni 4839   class class class wbr 5074  ran crn 5590  cima 5592  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268  Fun wfun 6427  cfv 6433  (class class class)co 7275   bday cbday 33845   <<s csslt 33975   |s cscut 33977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator