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Theorem noetalem4 33117
Description: Lemma for noeta 33119. Bound the birthday of 𝑍 above. (Contributed by Scott Fenton, 6-Dec-2021.)
Hypotheses
Ref Expression
noetalem.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
noetalem.2 𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))
Assertion
Ref Expression
noetalem4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
Distinct variable group:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑍(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetalem4
StepHypRef Expression
1 noetalem.1 . . . . . . 7 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
21nosupno 33100 . . . . . 6 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
3 bdayval 33052 . . . . . 6 (𝑆 No → ( bday 𝑆) = dom 𝑆)
42, 3syl 17 . . . . 5 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) = dom 𝑆)
51nosupbday 33102 . . . . 5 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) ⊆ suc ( bday 𝐴))
64, 5eqsstrrd 4003 . . . 4 ((𝐴 No 𝐴 ∈ V) → dom 𝑆 ⊆ suc ( bday 𝐴))
76adantr 481 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → dom 𝑆 ⊆ suc ( bday 𝐴))
8 unss1 4152 . . 3 (dom 𝑆 ⊆ suc ( bday 𝐴) → (dom 𝑆 ∪ suc ( bday 𝐵)) ⊆ (suc ( bday 𝐴) ∪ suc ( bday 𝐵)))
97, 8syl 17 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → (dom 𝑆 ∪ suc ( bday 𝐵)) ⊆ (suc ( bday 𝐴) ∪ suc ( bday 𝐵)))
10 simpll 763 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐴 No )
11 simplr 765 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐴 ∈ V)
12 simprr 769 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐵 ∈ V)
13 noetalem.2 . . . . . 6 𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))
141, 13noetalem1 33114 . . . . 5 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
1510, 11, 12, 14syl3anc 1363 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝑍 No )
16 bdayval 33052 . . . 4 (𝑍 No → ( bday 𝑍) = dom 𝑍)
1715, 16syl 17 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) = dom 𝑍)
1813dmeqi 5766 . . . 4 dom 𝑍 = dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))
19 dmun 5772 . . . . 5 dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))
20 1oex 8099 . . . . . . . . 9 1o ∈ V
2120snnz 4703 . . . . . . . 8 {1o} ≠ ∅
22 dmxp 5792 . . . . . . . 8 ({1o} ≠ ∅ → dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}) = (suc ( bday 𝐵) ∖ dom 𝑆))
2321, 22ax-mp 5 . . . . . . 7 dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}) = (suc ( bday 𝐵) ∖ dom 𝑆)
2423uneq2i 4133 . . . . . 6 (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ (suc ( bday 𝐵) ∖ dom 𝑆))
25 undif2 4421 . . . . . 6 (dom 𝑆 ∪ (suc ( bday 𝐵) ∖ dom 𝑆)) = (dom 𝑆 ∪ suc ( bday 𝐵))
2624, 25eqtri 2841 . . . . 5 (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ suc ( bday 𝐵))
2719, 26eqtri 2841 . . . 4 dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ suc ( bday 𝐵))
2818, 27eqtri 2841 . . 3 dom 𝑍 = (dom 𝑆 ∪ suc ( bday 𝐵))
2917, 28syl6eq 2869 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) = (dom 𝑆 ∪ suc ( bday 𝐵)))
30 imaundi 6001 . . . . . . 7 ( bday “ (𝐴𝐵)) = (( bday 𝐴) ∪ ( bday 𝐵))
3130unieqi 4839 . . . . . 6 ( bday “ (𝐴𝐵)) = (( bday 𝐴) ∪ ( bday 𝐵))
32 uniun 4849 . . . . . 6 (( bday 𝐴) ∪ ( bday 𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵))
3331, 32eqtri 2841 . . . . 5 ( bday “ (𝐴𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵))
34 suceq 6249 . . . . 5 ( ( bday “ (𝐴𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵)) → suc ( bday “ (𝐴𝐵)) = suc ( ( bday 𝐴) ∪ ( bday 𝐵)))
3533, 34ax-mp 5 . . . 4 suc ( bday “ (𝐴𝐵)) = suc ( ( bday 𝐴) ∪ ( bday 𝐵))
36 imassrn 5933 . . . . . . 7 ( bday 𝐴) ⊆ ran bday
37 bdayfo 33079 . . . . . . . 8 bday : No onto→On
38 forn 6586 . . . . . . . 8 ( bday : No onto→On → ran bday = On)
3937, 38ax-mp 5 . . . . . . 7 ran bday = On
4036, 39sseqtri 4000 . . . . . 6 ( bday 𝐴) ⊆ On
41 ssorduni 7489 . . . . . 6 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
4240, 41ax-mp 5 . . . . 5 Ord ( bday 𝐴)
43 imassrn 5933 . . . . . . 7 ( bday 𝐵) ⊆ ran bday
4443, 39sseqtri 4000 . . . . . 6 ( bday 𝐵) ⊆ On
45 ssorduni 7489 . . . . . 6 (( bday 𝐵) ⊆ On → Ord ( bday 𝐵))
4644, 45ax-mp 5 . . . . 5 Ord ( bday 𝐵)
47 ordsucun 7529 . . . . 5 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → suc ( ( bday 𝐴) ∪ ( bday 𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵)))
4842, 46, 47mp2an 688 . . . 4 suc ( ( bday 𝐴) ∪ ( bday 𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵))
4935, 48eqtri 2841 . . 3 suc ( bday “ (𝐴𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵))
5049a1i 11 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → suc ( bday “ (𝐴𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵)))
519, 29, 503sstr4d 4011 1 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  cun 3931  wss 3933  c0 4288  ifcif 4463  {csn 4557  cop 4563   cuni 4830   class class class wbr 5057  cmpt 5137   × cxp 5546  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Ord word 6183  Oncon0 6184  suc csuc 6186  cio 6305  ontowfo 6346  cfv 6348  crio 7102  1oc1o 8084  2oc2o 8085   No csur 33044   <s cslt 33045   bday cbday 33046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-1o 8091  df-2o 8092  df-no 33047  df-slt 33048  df-bday 33049
This theorem is referenced by:  noetalem5  33118
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