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Theorem noetalem4 32242
Description: Lemma for noeta 32244. 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 32225 . . . . . 6 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
3 bdayval 32177 . . . . . 6 (𝑆 No → ( bday 𝑆) = dom 𝑆)
42, 3syl 17 . . . . 5 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) = dom 𝑆)
51nosupbday 32227 . . . . 5 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) ⊆ suc ( bday 𝐴))
64, 5eqsstr3d 3800 . . . 4 ((𝐴 No 𝐴 ∈ V) → dom 𝑆 ⊆ suc ( bday 𝐴))
76adantr 472 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → dom 𝑆 ⊆ suc ( bday 𝐴))
8 unss1 3944 . . 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 783 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐴 No )
11 simplr 785 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐴 ∈ V)
12 simprr 789 . . . . 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 32239 . . . . 5 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
1510, 11, 12, 14syl3anc 1490 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝑍 No )
16 bdayval 32177 . . . 4 (𝑍 No → ( bday 𝑍) = dom 𝑍)
1715, 16syl 17 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) = dom 𝑍)
1813dmeqi 5493 . . . 4 dom 𝑍 = dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
19 dmun 5499 . . . . 5 dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
20 1oex 7772 . . . . . . . . 9 1𝑜 ∈ V
2120snnz 4463 . . . . . . . 8 {1𝑜} ≠ ∅
22 dmxp 5512 . . . . . . . 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 3926 . . . . . 6 (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ (suc ( bday 𝐵) ∖ dom 𝑆))
25 undif2 4204 . . . . . 6 (dom 𝑆 ∪ (suc ( bday 𝐵) ∖ dom 𝑆)) = (dom 𝑆 ∪ suc ( bday 𝐵))
2624, 25eqtri 2787 . . . . 5 (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ suc ( bday 𝐵))
2719, 26eqtri 2787 . . . 4 dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ suc ( bday 𝐵))
2818, 27eqtri 2787 . . 3 dom 𝑍 = (dom 𝑆 ∪ suc ( bday 𝐵))
2917, 28syl6eq 2815 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) = (dom 𝑆 ∪ suc ( bday 𝐵)))
30 imaundi 5728 . . . . . . 7 ( bday “ (𝐴𝐵)) = (( bday 𝐴) ∪ ( bday 𝐵))
3130unieqi 4603 . . . . . 6 ( bday “ (𝐴𝐵)) = (( bday 𝐴) ∪ ( bday 𝐵))
32 uniun 4615 . . . . . 6 (( bday 𝐴) ∪ ( bday 𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵))
3331, 32eqtri 2787 . . . . 5 ( bday “ (𝐴𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵))
34 suceq 5973 . . . . 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 5659 . . . . . . 7 ( bday 𝐴) ⊆ ran bday
37 bdayfo 32204 . . . . . . . 8 bday : No onto→On
38 forn 6301 . . . . . . . 8 ( bday : No onto→On → ran bday = On)
3937, 38ax-mp 5 . . . . . . 7 ran bday = On
4036, 39sseqtri 3797 . . . . . 6 ( bday 𝐴) ⊆ On
41 ssorduni 7183 . . . . . 6 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
4240, 41ax-mp 5 . . . . 5 Ord ( bday 𝐴)
43 imassrn 5659 . . . . . . 7 ( bday 𝐵) ⊆ ran bday
4443, 39sseqtri 3797 . . . . . 6 ( bday 𝐵) ⊆ On
45 ssorduni 7183 . . . . . 6 (( bday 𝐵) ⊆ On → Ord ( bday 𝐵))
4644, 45ax-mp 5 . . . . 5 Ord ( bday 𝐵)
47 ordsucun 7223 . . . . 5 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → suc ( ( bday 𝐴) ∪ ( bday 𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵)))
4842, 46, 47mp2an 683 . . . 4 suc ( ( bday 𝐴) ∪ ( bday 𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵))
4935, 48eqtri 2787 . . 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 3808 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 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  wss 3732  c0 4079  ifcif 4243  {csn 4334  cop 4340   cuni 4594   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  Ord word 5907  Oncon0 5908  suc csuc 5910  cio 6029  ontowfo 6066  cfv 6068  crio 6802  1𝑜c1o 7757  2𝑜c2o 7758   No csur 32169   <s cslt 32170   bday cbday 32171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-1o 7764  df-2o 7765  df-no 32172  df-slt 32173  df-bday 32174
This theorem is referenced by:  noetalem5  32243
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