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Theorem scutbdaybnd2lim 27556
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 27555 . . . 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 27511 . . . . . . . . 9 Fun bday
4 ssltex1 27525 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
5 ssltex2 27526 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐵 ∈ V)
6 unexg 7740 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
74, 5, 6syl2anc 583 . . . . . . . . 9 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
8 funimaexg 6634 . . . . . . . . 9 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
93, 7, 8sylancr 586 . . . . . . . 8 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
109uniexd 7736 . . . . . . 7 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
1110adantr 480 . . . . . 6 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday “ (𝐴𝐵)) ∈ V)
12 nlimsucg 7835 . . . . . 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 6376 . . . . . . 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 2946 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵)))
19 bdayelon 27515 . . . . 5 ( bday ‘(𝐴 |s 𝐵)) ∈ On
2019onordi 6475 . . . 4 Ord ( bday ‘(𝐴 |s 𝐵))
21 imassrn 6070 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
22 bdayrn 27514 . . . . . . 7 ran bday = On
2321, 22sseqtri 4018 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
24 ssorduni 7770 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
2523, 24ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
26 ordsuc 7805 . . . . 5 (Ord ( bday “ (𝐴𝐵)) ↔ Ord suc ( bday “ (𝐴𝐵)))
2725, 26mpbi 229 . . . 4 Ord suc ( bday “ (𝐴𝐵))
28 ordelssne 6391 . . . 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 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 6453 . . 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 1540  wcel 2105  wne 2939  Vcvv 3473  cun 3946  wss 3948   cuni 4908   class class class wbr 5148  ran crn 5677  cima 5679  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  Fun wfun 6537  cfv 6543  (class class class)co 7412   bday cbday 27382   <<s csslt 27519   |s cscut 27521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1o 8470  df-2o 8471  df-no 27383  df-slt 27384  df-bday 27385  df-sslt 27520  df-scut 27522
This theorem is referenced by: (None)
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