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Theorem scutbdaybnd2lim 27737
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 27736 . . . 4 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21adantr 480 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
3 bdayfun 27692 . . . . . . . . 9 Fun bday
4 ssltex1 27706 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
5 ssltex2 27707 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐵 ∈ V)
6 unexg 7745 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
74, 5, 6syl2anc 583 . . . . . . . . 9 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
8 funimaexg 6633 . . . . . . . . 9 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
93, 7, 8sylancr 586 . . . . . . . 8 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
109uniexd 7741 . . . . . . 7 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
1110adantr 480 . . . . . 6 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday “ (𝐴𝐵)) ∈ V)
12 nlimsucg 7840 . . . . . 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 6375 . . . . . . 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 481 . . . . 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 2942 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵)))
19 bdayelon 27696 . . . . 5 ( bday ‘(𝐴 |s 𝐵)) ∈ On
2019onordi 6474 . . . 4 Ord ( bday ‘(𝐴 |s 𝐵))
21 imassrn 6068 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
22 bdayrn 27695 . . . . . . 7 ran bday = On
2321, 22sseqtri 4014 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
24 ssorduni 7775 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
2523, 24ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
26 ordsuc 7810 . . . . 5 (Ord ( bday “ (𝐴𝐵)) ↔ Ord suc ( bday “ (𝐴𝐵)))
2725, 26mpbi 229 . . . 4 Ord suc ( bday “ (𝐴𝐵))
28 ordelssne 6390 . . . 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 582 . 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 6452 . . 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 585 . 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 395   = wceq 1534  wcel 2099  wne 2935  Vcvv 3469  cun 3942  wss 3944   cuni 4903   class class class wbr 5142  ran crn 5673  cima 5675  Ord word 6362  Oncon0 6363  Lim wlim 6364  suc csuc 6365  Fun wfun 6536  cfv 6542  (class class class)co 7414   bday cbday 27562   <<s csslt 27700   |s cscut 27702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1o 8480  df-2o 8481  df-no 27563  df-slt 27564  df-bday 27565  df-sslt 27701  df-scut 27703
This theorem is referenced by: (None)
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