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 33656
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 33655 . . . 4 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21adantr 484 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
3 bdayfun 33612 . . . . . . . . 9 Fun bday
4 ssltex1 33626 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
5 ssltex2 33627 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐵 ∈ V)
6 unexg 7492 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
74, 5, 6syl2anc 587 . . . . . . . . 9 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
8 funimaexg 6425 . . . . . . . . 9 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
93, 7, 8sylancr 590 . . . . . . . 8 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
109uniexd 7488 . . . . . . 7 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
1110adantr 484 . . . . . 6 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday “ (𝐴𝐵)) ∈ V)
12 nlimsucg 7578 . . . . . 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 6184 . . . . . . 7 (( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)) → (Lim ( bday ‘(𝐴 |s 𝐵)) ↔ Lim suc ( bday “ (𝐴𝐵))))
1514biimpcd 252 . . . . . 6 (Lim ( bday ‘(𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)) → Lim suc ( bday “ (𝐴𝐵))))
1615adantl 485 . . . . 5 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → (( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)) → Lim suc ( bday “ (𝐴𝐵))))
1713, 16mtod 201 . . . 4 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ¬ ( bday ‘(𝐴 |s 𝐵)) = suc ( bday “ (𝐴𝐵)))
1817neqned 2941 . . 3 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ( bday “ (𝐴𝐵)))
19 bdayelon 33616 . . . . 5 ( bday ‘(𝐴 |s 𝐵)) ∈ On
2019onordi 6277 . . . 4 Ord ( bday ‘(𝐴 |s 𝐵))
21 imassrn 5914 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
22 bdayrn 33615 . . . . . . 7 ran bday = On
2321, 22sseqtri 3913 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
24 ssorduni 7521 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
2523, 24ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
26 ordsuc 7550 . . . . 5 (Ord ( bday “ (𝐴𝐵)) ↔ Ord suc ( bday “ (𝐴𝐵)))
2725, 26mpbi 233 . . . 4 Ord suc ( bday “ (𝐴𝐵))
28 ordelssne 6199 . . . 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 692 . . 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 586 . 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 6258 . . 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 589 . 2 ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ (𝐴𝐵)) ↔ ( bday ‘(𝐴 |s 𝐵)) ∈ suc ( bday “ (𝐴𝐵))))
3430, 33mpbird 260 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 209  wa 399   = wceq 1542  wcel 2114  wne 2934  Vcvv 3398  cun 3841  wss 3843   cuni 4796   class class class wbr 5030  ran crn 5526  cima 5528  Ord word 6171  Oncon0 6172  Lim wlim 6173  suc csuc 6174  Fun wfun 6333  cfv 6339  (class class class)co 7172   bday cbday 33490   <<s csslt 33620   |s cscut 33622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1o 8133  df-2o 8134  df-no 33491  df-slt 33492  df-bday 33493  df-sslt 33621  df-scut 33623
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator