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Theorem noeta2 33624
Description: A version of noeta 33591 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 33525 . . . . . . 7 bday : No onto→On
3 fofun 6593 . . . . . . 7 ( bday : No onto→On → Fun bday )
42, 3ax-mp 5 . . . . . 6 Fun bday
5 simp1r 1199 . . . . . . 7 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝐴𝑉)
6 simp2r 1201 . . . . . . 7 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → 𝐵𝑊)
7 unexg 7492 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
85, 6, 7syl2anc 587 . . . . . 6 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → (𝐴𝐵) ∈ V)
9 funimaexg 6425 . . . . . 6 ((Fun bday ∧ (𝐴𝐵) ∈ V) → ( bday “ (𝐴𝐵)) ∈ V)
104, 8, 9sylancr 590 . . . . 5 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ∈ V)
1110uniexd 7488 . . . 4 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ∈ V)
12 imassrn 5914 . . . . . . 7 ( bday “ (𝐴𝐵)) ⊆ ran bday
13 forn 6595 . . . . . . . 8 ( bday : No onto→On → ran bday = On)
142, 13ax-mp 5 . . . . . . 7 ran bday = On
1512, 14sseqtri 3913 . . . . . 6 ( bday “ (𝐴𝐵)) ⊆ On
16 ssorduni 7521 . . . . . 6 (( bday “ (𝐴𝐵)) ⊆ On → Ord ( bday “ (𝐴𝐵)))
1715, 16ax-mp 5 . . . . 5 Ord ( bday “ (𝐴𝐵))
18 elon2 6183 . . . . 5 ( ( bday “ (𝐴𝐵)) ∈ On ↔ (Ord ( bday “ (𝐴𝐵)) ∧ ( bday “ (𝐴𝐵)) ∈ V))
1917, 18mpbiran 709 . . . 4 ( ( bday “ (𝐴𝐵)) ∈ On ↔ ( bday “ (𝐴𝐵)) ∈ V)
2011, 19sylibr 237 . . 3 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴𝐵)) ∈ On)
21 sucelon 7553 . . 3 ( ( bday “ (𝐴𝐵)) ∈ On ↔ suc ( bday “ (𝐴𝐵)) ∈ On)
2220, 21sylib 221 . 2 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → suc ( bday “ (𝐴𝐵)) ∈ On)
23 onsucuni 7564 . . 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 33591 . 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 836 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 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wrex 3054  Vcvv 3398  cun 3841  wss 3843   cuni 4796   class class class wbr 5030  ran crn 5526  cima 5528  Ord word 6171  Oncon0 6172  suc csuc 6174  Fun wfun 6333  ontowfo 6337  cfv 6339   No csur 33488   <s cslt 33489   bday cbday 33490
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-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-1o 8133  df-2o 8134  df-no 33491  df-slt 33492  df-bday 33493
This theorem is referenced by:  conway  33638
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