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Theorem etasslt2 27859
Description: A version of etasslt 27858 with fewer hypotheses but a weaker upper bound. (Contributed by Scott Fenton, 10-Dec-2021.)
Assertion
Ref Expression
etasslt2 (𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem etasslt2
StepHypRef Expression
1 bdayfun 27817 . . . . . 6 Fun bday
2 ssltex1 27831 . . . . . . 7 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltex2 27832 . . . . . . 7 (𝐴 <<s 𝐵𝐵 ∈ V)
4 unexg 7763 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
52, 3, 4syl2anc 584 . . . . . 6 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
6 funimaexg 6653 . . . . . 6 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
71, 5, 6sylancr 587 . . . . 5 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
87uniexd 7762 . . . 4 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
9 imassrn 6089 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
10 bdayrn 27820 . . . . . . 7 ran bday = On
119, 10sseqtri 4032 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
12 ssorduni 7799 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
1311, 12ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
14 elon2 6395 . . . . 5 ( ( bday “ (𝐴𝐵)) ∈ On ↔ (Ord ( bday “ (𝐴𝐵)) ∧ ( bday “ (𝐴𝐵)) ∈ V))
1513, 14mpbiran 709 . . . 4 ( ( bday “ (𝐴𝐵)) ∈ On ↔ ( bday “ (𝐴𝐵)) ∈ V)
168, 15sylibr 234 . . 3 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ On)
17 onsucb 7837 . . 3 ( ( bday “ (𝐴𝐵)) ∈ On ↔ suc ( bday “ (𝐴𝐵)) ∈ On)
1816, 17sylib 218 . 2 (𝐴 <<s 𝐵 → suc ( bday “ (𝐴𝐵)) ∈ On)
19 onsucuni 7848 . . 3 (( bday “ (𝐴𝐵)) ⊆ On → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
2011, 19mp1i 13 . 2 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21 etasslt 27858 . 2 ((𝐴 <<s 𝐵 ∧ suc ( bday “ (𝐴𝐵)) ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵))) → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
2218, 20, 21mpd3an23 1465 1 (𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2108  wrex 3070  Vcvv 3480  cun 3949  wss 3951  {csn 4626   cuni 4907   class class class wbr 5143  ran crn 5686  cima 5688  Ord word 6383  Oncon0 6384  suc csuc 6386  Fun wfun 6555  cfv 6561   No csur 27684   bday cbday 27686   <<s csslt 27825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826
This theorem is referenced by:  scutbdaybnd2  27861
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