Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  noetalem4 Structured version   Visualization version   GIF version

Theorem noetalem4 32203
Description: Lemma for noeta 32205. 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 32186 . . . . . 6 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
3 bdayval 32138 . . . . . 6 (𝑆 No → ( bday 𝑆) = dom 𝑆)
42, 3syl 17 . . . . 5 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) = dom 𝑆)
51nosupbday 32188 . . . . 5 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) ⊆ suc ( bday 𝐴))
64, 5eqsstr3d 3789 . . . 4 ((𝐴 No 𝐴 ∈ V) → dom 𝑆 ⊆ suc ( bday 𝐴))
76adantr 466 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → dom 𝑆 ⊆ suc ( bday 𝐴))
8 unss1 3933 . . 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 750 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐴 No )
11 simplr 752 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝐴 ∈ V)
12 simprr 756 . . . . 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 32200 . . . . 5 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
1510, 11, 12, 14syl3anc 1476 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → 𝑍 No )
16 bdayval 32138 . . . 4 (𝑍 No → ( bday 𝑍) = dom 𝑍)
1715, 16syl 17 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) = dom 𝑍)
1813dmeqi 5463 . . . 4 dom 𝑍 = dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
19 dmun 5469 . . . . 5 dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
20 1oex 7721 . . . . . . . . 9 1𝑜 ∈ V
2120snnz 4444 . . . . . . . 8 {1𝑜} ≠ ∅
22 dmxp 5482 . . . . . . . 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 3915 . . . . . 6 (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ (suc ( bday 𝐵) ∖ dom 𝑆))
25 undif2 4186 . . . . . 6 (dom 𝑆 ∪ (suc ( bday 𝐵) ∖ dom 𝑆)) = (dom 𝑆 ∪ suc ( bday 𝐵))
2624, 25eqtri 2793 . . . . 5 (dom 𝑆 ∪ dom ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ suc ( bday 𝐵))
2719, 26eqtri 2793 . . . 4 dom (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ suc ( bday 𝐵))
2818, 27eqtri 2793 . . 3 dom 𝑍 = (dom 𝑆 ∪ suc ( bday 𝐵))
2917, 28syl6eq 2821 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) = (dom 𝑆 ∪ suc ( bday 𝐵)))
30 imaundi 5686 . . . . . . 7 ( bday “ (𝐴𝐵)) = (( bday 𝐴) ∪ ( bday 𝐵))
3130unieqi 4583 . . . . . 6 ( bday “ (𝐴𝐵)) = (( bday 𝐴) ∪ ( bday 𝐵))
32 uniun 4593 . . . . . 6 (( bday 𝐴) ∪ ( bday 𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵))
3331, 32eqtri 2793 . . . . 5 ( bday “ (𝐴𝐵)) = ( ( bday 𝐴) ∪ ( bday 𝐵))
34 suceq 5933 . . . . 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 5618 . . . . . . 7 ( bday 𝐴) ⊆ ran bday
37 bdayfo 32165 . . . . . . . 8 bday : No onto→On
38 forn 6259 . . . . . . . 8 ( bday : No onto→On → ran bday = On)
3937, 38ax-mp 5 . . . . . . 7 ran bday = On
4036, 39sseqtri 3786 . . . . . 6 ( bday 𝐴) ⊆ On
41 ssorduni 7132 . . . . . 6 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
4240, 41ax-mp 5 . . . . 5 Ord ( bday 𝐴)
43 imassrn 5618 . . . . . . 7 ( bday 𝐵) ⊆ ran bday
4443, 39sseqtri 3786 . . . . . 6 ( bday 𝐵) ⊆ On
45 ssorduni 7132 . . . . . 6 (( bday 𝐵) ⊆ On → Ord ( bday 𝐵))
4644, 45ax-mp 5 . . . . 5 Ord ( bday 𝐵)
47 ordsucun 7172 . . . . 5 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → suc ( ( bday 𝐴) ∪ ( bday 𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵)))
4842, 46, 47mp2an 672 . . . 4 suc ( ( bday 𝐴) ∪ ( bday 𝐵)) = (suc ( bday 𝐴) ∪ suc ( bday 𝐵))
4935, 48eqtri 2793 . . 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 3797 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 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  cun 3721  wss 3723  c0 4063  ifcif 4225  {csn 4316  cop 4322   cuni 4574   class class class wbr 4786  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Ord word 5865  Oncon0 5866  suc csuc 5868  cio 5992  ontowfo 6029  cfv 6031  crio 6753  1𝑜c1o 7706  2𝑜c2o 7707   No csur 32130   <s cslt 32131   bday cbday 32132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-1o 7713  df-2o 7714  df-no 32133  df-slt 32134  df-bday 32135
This theorem is referenced by:  noetalem5  32204
  Copyright terms: Public domain W3C validator