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Theorem etasslt2 27175
Description: A version of etasslt 27174 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 27134 . . . . . 6 Fun bday
2 ssltex1 27148 . . . . . . 7 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltex2 27149 . . . . . . 7 (𝐴 <<s 𝐵𝐵 ∈ V)
4 unexg 7684 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
52, 3, 4syl2anc 585 . . . . . 6 (𝐴 <<s 𝐵 → (𝐴𝐵) ∈ V)
6 funimaexg 6588 . . . . . 6 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
71, 5, 6sylancr 588 . . . . 5 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ∈ V)
87uniexd 7680 . . . 4 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ V)
9 imassrn 6025 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
10 bdayrn 27137 . . . . . . 7 ran bday = On
119, 10sseqtri 3981 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
12 ssorduni 7714 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
1311, 12ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
14 elon2 6329 . . . . 5 ( ( bday “ (𝐴𝐵)) ∈ On ↔ (Ord ( bday “ (𝐴𝐵)) ∧ ( bday “ (𝐴𝐵)) ∈ V))
1513, 14mpbiran 708 . . . 4 ( ( bday “ (𝐴𝐵)) ∈ On ↔ ( bday “ (𝐴𝐵)) ∈ V)
168, 15sylibr 233 . . 3 (𝐴 <<s 𝐵 ( bday “ (𝐴𝐵)) ∈ On)
17 onsucb 7753 . . 3 ( ( bday “ (𝐴𝐵)) ∈ On ↔ suc ( bday “ (𝐴𝐵)) ∈ On)
1816, 17sylib 217 . 2 (𝐴 <<s 𝐵 → suc ( bday “ (𝐴𝐵)) ∈ On)
19 onsucuni 7764 . . 3 (( bday “ (𝐴𝐵)) ⊆ On → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
2011, 19mp1i 13 . 2 (𝐴 <<s 𝐵 → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
21 etasslt 27174 . 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 1464 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 1088  wcel 2107  wrex 3070  Vcvv 3444  cun 3909  wss 3911  {csn 4587   cuni 4866   class class class wbr 5106  ran crn 5635  cima 5637  Ord word 6317  Oncon0 6318  suc csuc 6320  Fun wfun 6491  cfv 6497   No csur 27004   bday cbday 27006   <<s csslt 27142
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 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143
This theorem is referenced by:  scutbdaybnd2  27177
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