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Theorem nobndlem8 31098
Description: Lemma for nobndup 31099 and nobnddown 31100. Bound the birthday of (𝐶 × {𝑆}) above. (Contributed by Scott Fenton, 10-Apr-2017.)
Hypotheses
Ref Expression
nobndlem8.1 𝑆 ∈ {1𝑜, 2𝑜}
nobndlem8.2 𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆}
Assertion
Ref Expression
nobndlem8 ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))
Distinct variable groups:   𝐹,𝑎,𝑏,𝑛   𝑆,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑛,𝑎,𝑏)   𝐶(𝑛,𝑎,𝑏)   𝑆(𝑛)

Proof of Theorem nobndlem8
StepHypRef Expression
1 elex 3185 . 2 (𝐹𝐴𝐹 ∈ V)
2 nobndlem8.1 . . . . 5 𝑆 ∈ {1𝑜, 2𝑜}
3 nobndlem8.2 . . . . 5 𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆}
42, 3nobndlem3 31093 . . . 4 ((𝐹 No 𝐹 ∈ V) → ( bday ‘(𝐶 × {𝑆})) = 𝐶)
54, 3syl6eq 2660 . . 3 ((𝐹 No 𝐹 ∈ V) → ( bday ‘(𝐶 × {𝑆})) = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆})
6 nobndlem1 31091 . . . . 5 (𝐹 ∈ V → suc ( bday 𝐹) ∈ On)
76adantl 481 . . . 4 ((𝐹 No 𝐹 ∈ V) → suc ( bday 𝐹) ∈ On)
8 bdayfn 31078 . . . . . . . . . . 11 bday Fn No
9 fnfvima 6400 . . . . . . . . . . 11 (( bday Fn No 𝐹 No 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
108, 9mp3an1 1403 . . . . . . . . . 10 ((𝐹 No 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
11103adant2 1073 . . . . . . . . 9 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
12 elssuni 4403 . . . . . . . . 9 (( bday 𝑛) ∈ ( bday 𝐹) → ( bday 𝑛) ⊆ ( bday 𝐹))
1311, 12syl 17 . . . . . . . 8 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝑛) ⊆ ( bday 𝐹))
14 bdayelon 31079 . . . . . . . . 9 ( bday 𝑛) ∈ On
15 sucelon 6909 . . . . . . . . . . 11 ( ( bday 𝐹) ∈ On ↔ suc ( bday 𝐹) ∈ On)
166, 15sylibr 223 . . . . . . . . . 10 (𝐹 ∈ V → ( bday 𝐹) ∈ On)
17163ad2ant2 1076 . . . . . . . . 9 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝐹) ∈ On)
18 onsssuc 5730 . . . . . . . . 9 ((( bday 𝑛) ∈ On ∧ ( bday 𝐹) ∈ On) → (( bday 𝑛) ⊆ ( bday 𝐹) ↔ ( bday 𝑛) ∈ suc ( bday 𝐹)))
1914, 17, 18sylancr 694 . . . . . . . 8 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → (( bday 𝑛) ⊆ ( bday 𝐹) ↔ ( bday 𝑛) ∈ suc ( bday 𝐹)))
2013, 19mpbid 221 . . . . . . 7 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝑛) ∈ suc ( bday 𝐹))
21 ssel2 3563 . . . . . . . . 9 ((𝐹 No 𝑛𝐹) → 𝑛 No )
22 fvnobday 31081 . . . . . . . . . 10 (𝑛 No → (𝑛‘( bday 𝑛)) = ∅)
232nosgnn0i 31056 . . . . . . . . . . 11 ∅ ≠ 𝑆
2423a1i 11 . . . . . . . . . 10 (𝑛 No → ∅ ≠ 𝑆)
2522, 24eqnetrd 2849 . . . . . . . . 9 (𝑛 No → (𝑛‘( bday 𝑛)) ≠ 𝑆)
2621, 25syl 17 . . . . . . . 8 ((𝐹 No 𝑛𝐹) → (𝑛‘( bday 𝑛)) ≠ 𝑆)
27263adant2 1073 . . . . . . 7 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → (𝑛‘( bday 𝑛)) ≠ 𝑆)
28 fveq2 6103 . . . . . . . . 9 (𝑏 = ( bday 𝑛) → (𝑛𝑏) = (𝑛‘( bday 𝑛)))
2928neeq1d 2841 . . . . . . . 8 (𝑏 = ( bday 𝑛) → ((𝑛𝑏) ≠ 𝑆 ↔ (𝑛‘( bday 𝑛)) ≠ 𝑆))
3029rspcev 3282 . . . . . . 7 ((( bday 𝑛) ∈ suc ( bday 𝐹) ∧ (𝑛‘( bday 𝑛)) ≠ 𝑆) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
3120, 27, 30syl2anc 691 . . . . . 6 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
32313expa 1257 . . . . 5 (((𝐹 No 𝐹 ∈ V) ∧ 𝑛𝐹) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
3332ralrimiva 2949 . . . 4 ((𝐹 No 𝐹 ∈ V) → ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
34 rexeq 3116 . . . . . 6 (𝑎 = suc ( bday 𝐹) → (∃𝑏𝑎 (𝑛𝑏) ≠ 𝑆 ↔ ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆))
3534ralbidv 2969 . . . . 5 (𝑎 = suc ( bday 𝐹) → (∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆 ↔ ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆))
3635intminss 4438 . . . 4 ((suc ( bday 𝐹) ∈ On ∧ ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆) → {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆} ⊆ suc ( bday 𝐹))
377, 33, 36syl2anc 691 . . 3 ((𝐹 No 𝐹 ∈ V) → {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆} ⊆ suc ( bday 𝐹))
385, 37eqsstrd 3602 . 2 ((𝐹 No 𝐹 ∈ V) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))
391, 38sylan2 490 1 ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  wss 3540  c0 3874  {csn 4125  {cpr 4127   cuni 4372   cint 4410   × cxp 5036  cima 5041  Oncon0 5640  suc csuc 5642   Fn wfn 5799  cfv 5804  1𝑜c1o 7440  2𝑜c2o 7441   No csur 31037   bday cbday 31039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-2o 7448  df-no 31040  df-bday 31042
This theorem is referenced by:  nobndup  31099  nobnddown  31100
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