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Theorem noeta2 33979
Description: A version of noeta 33946 with fewer hypotheses but a weaker upper bound (Contributed by Scott Fenton, 7-Dec-2021.)
Assertion
Ref Expression
noeta2 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem noeta2
StepHypRef Expression
1 id 22 . 2 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦))
2 bdayfo 33880 . . . . . . 7 bday : No onto→On
3 fofun 6689 . . . . . . 7 ( bday : No onto→On → Fun bday )
42, 3ax-mp 5 . . . . . 6 Fun bday
5 simp1r 1197 . . . . . . 7 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝐴𝑉)
6 simp2r 1199 . . . . . . 7 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝐵𝑊)
7 unexg 7599 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
85, 6, 7syl2anc 584 . . . . . 6 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → (𝐴𝐵) ∈ V)
9 funimaexg 6520 . . . . . 6 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
104, 8, 9sylancr 587 . . . . 5 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ∈ V)
1110uniexd 7595 . . . 4 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ∈ V)
12 imassrn 5980 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
13 forn 6691 . . . . . . . 8 ( bday : No onto→On → ran bday = On)
142, 13ax-mp 5 . . . . . . 7 ran bday = On
1512, 14sseqtri 3957 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
16 ssorduni 7629 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
1715, 16ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
18 elon2 6277 . . . . 5 ( ( bday “ (𝐴𝐵)) ∈ On ↔ (Ord ( bday “ (𝐴𝐵)) ∧ ( bday “ (𝐴𝐵)) ∈ V))
1917, 18mpbiran 706 . . . 4 ( ( bday “ (𝐴𝐵)) ∈ On ↔ ( bday “ (𝐴𝐵)) ∈ V)
2011, 19sylibr 233 . . 3 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ∈ On)
21 sucelon 7664 . . 3 ( ( bday “ (𝐴𝐵)) ∈ On ↔ suc ( bday “ (𝐴𝐵)) ∈ On)
2220, 21sylib 217 . 2 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → suc ( bday “ (𝐴𝐵)) ∈ On)
23 onsucuni 7675 . . 3 (( bday “ (𝐴𝐵)) ⊆ On → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
2415, 23mp1i 13 . 2 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
25 noeta 33946 . 2 ((((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) ∧ (suc ( bday “ (𝐴𝐵)) ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
261, 22, 24, 25syl12anc 834 1 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  cun 3885  wss 3887   cuni 4839   class class class wbr 5074  ran crn 5590  cima 5592  Ord word 6265  Oncon0 6266  suc csuc 6268  Fun wfun 6427  ontowfo 6431  cfv 6433   No csur 33843   <s cslt 33844   bday cbday 33845
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
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 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848
This theorem is referenced by:  conway  33993
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