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Theorem scutbdaybnd2lim 27318
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 27317 . . . 4 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21adantr 482 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
3 bdayfun 27274 . . . . . . . . 9 Fun bday
4 ssltex1 27288 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
5 ssltex2 27289 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐵 ∈ V)
6 unexg 7736 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
74, 5, 6syl2anc 585 . . . . . . . . 9 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
8 funimaexg 6635 . . . . . . . . 9 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
93, 7, 8sylancr 588 . . . . . . . 8 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
109uniexd 7732 . . . . . . 7 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
1110adantr 482 . . . . . 6 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday “ (𝐴𝐵)) ∈ V)
12 nlimsucg 7831 . . . . . 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 6377 . . . . . . 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 483 . . . . 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 2948 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵)))
19 bdayelon 27278 . . . . 5 ( bday ‘(𝐴 |s 𝐵)) ∈ On
2019onordi 6476 . . . 4 Ord ( bday ‘(𝐴 |s 𝐵))
21 imassrn 6071 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
22 bdayrn 27277 . . . . . . 7 ran bday = On
2321, 22sseqtri 4019 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
24 ssorduni 7766 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
2523, 24ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
26 ordsuc 7801 . . . . 5 (Ord ( bday “ (𝐴𝐵)) ↔ Ord suc ( bday “ (𝐴𝐵)))
2725, 26mpbi 229 . . . 4 Ord suc ( bday “ (𝐴𝐵))
28 ordelssne 6392 . . . 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 691 . . 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 584 . 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 6454 . . 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 587 . 2 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ (𝐴𝐵)) ↔ ( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵))))
3430, 33mpbird 257 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 397   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cun 3947  wss 3949   cuni 4909   class class class wbr 5149  ran crn 5678  cima 5680  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  Fun wfun 6538  cfv 6544  (class class class)co 7409   bday cbday 27145   <<s csslt 27282   |s cscut 27284
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147  df-bday 27148  df-sslt 27283  df-scut 27285
This theorem is referenced by: (None)
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