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Theorem etasslt2 34017
Description: A version of etasslt 34016 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 33976 . . . . . 6 Fun bday
2 ssltex1 33990 . . . . . . 7 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltex2 33991 . . . . . . 7 (𝐴 <<s 𝐵𝐵 ∈ V)
4 unexg 7608 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
52, 3, 4syl2anc 584 . . . . . 6 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
6 funimaexg 6527 . . . . . 6 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
71, 5, 6sylancr 587 . . . . 5 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
87uniexd 7604 . . . 4 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
9 imassrn 5983 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
10 bdayrn 33979 . . . . . . 7 ran bday = On
119, 10sseqtri 3958 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
12 ssorduni 7638 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
1311, 12ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
14 elon2 6281 . . . . 5 ( ( bday “ (𝐴𝐵)) ∈ On ↔ (Ord ( bday “ (𝐴𝐵)) ∧ ( bday “ (𝐴𝐵)) ∈ V))
1513, 14mpbiran 706 . . . 4 ( ( bday “ (𝐴𝐵)) ∈ On ↔ ( bday “ (𝐴𝐵)) ∈ V)
168, 15sylibr 233 . . 3 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ On)
17 sucelon 7673 . . 3 ( ( bday “ (𝐴𝐵)) ∈ On ↔ suc ( bday “ (𝐴𝐵)) ∈ On)
1816, 17sylib 217 . 2 (𝐴 <<s 𝐵 → suc ( bday “ (𝐴𝐵)) ∈ On)
19 onsucuni 7684 . . 3 (( bday “ (𝐴𝐵)) ⊆ On → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
2011, 19mp1i 13 . 2 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21 etasslt 34016 . 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 1462 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 1086  wcel 2107  wrex 3066  Vcvv 3433  cun 3886  wss 3888  {csn 4562   cuni 4840   class class class wbr 5075  ran crn 5591  cima 5593  Ord word 6269  Oncon0 6270  suc csuc 6272  Fun wfun 6431  cfv 6437   No csur 33852   bday cbday 33854   <<s csslt 33984
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 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pr 5353  ax-un 7597
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6273  df-on 6274  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-1o 8306  df-2o 8307  df-no 33855  df-slt 33856  df-bday 33857  df-sslt 33985
This theorem is referenced by:  scutbdaybnd2  34019
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