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Theorem nobndlem8 31562
 Description: Lemma for nobndup 31563 and nobnddown 31564. 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 3198 . 2 (𝐹𝐴𝐹 ∈ V)
2 nobndlem8.1 . . . . 5 𝑆 ∈ {1𝑜, 2𝑜}
3 nobndlem8.2 . . . . 5 𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆}
42, 3nobndlem3 31557 . . . 4 ((𝐹 No 𝐹 ∈ V) → ( bday ‘(𝐶 × {𝑆})) = 𝐶)
54, 3syl6eq 2671 . . 3 ((𝐹 No 𝐹 ∈ V) → ( bday ‘(𝐶 × {𝑆})) = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆})
6 nobndlem1 31555 . . . . 5 (𝐹 ∈ V → suc ( bday 𝐹) ∈ On)
76adantl 482 . . . 4 ((𝐹 No 𝐹 ∈ V) → suc ( bday 𝐹) ∈ On)
8 bdayfn 31542 . . . . . . . . . . 11 bday Fn No
9 fnfvima 6450 . . . . . . . . . . 11 (( bday Fn No 𝐹 No 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
108, 9mp3an1 1408 . . . . . . . . . 10 ((𝐹 No 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
11103adant2 1078 . . . . . . . . 9 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
12 elssuni 4433 . . . . . . . . 9 (( bday 𝑛) ∈ ( bday 𝐹) → ( bday 𝑛) ⊆ ( bday 𝐹))
1311, 12syl 17 . . . . . . . 8 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝑛) ⊆ ( bday 𝐹))
14 bdayelon 31543 . . . . . . . . 9 ( bday 𝑛) ∈ On
15 sucelon 6964 . . . . . . . . . . 11 ( ( bday 𝐹) ∈ On ↔ suc ( bday 𝐹) ∈ On)
166, 15sylibr 224 . . . . . . . . . 10 (𝐹 ∈ V → ( bday 𝐹) ∈ On)
17163ad2ant2 1081 . . . . . . . . 9 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝐹) ∈ On)
18 onsssuc 5772 . . . . . . . . 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 222 . . . . . . 7 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ( bday 𝑛) ∈ suc ( bday 𝐹))
21 ssel2 3578 . . . . . . . . 9 ((𝐹 No 𝑛𝐹) → 𝑛 No )
22 fvnobday 31545 . . . . . . . . . 10 (𝑛 No → (𝑛‘( bday 𝑛)) = ∅)
232nosgnn0i 31513 . . . . . . . . . . 11 ∅ ≠ 𝑆
2423a1i 11 . . . . . . . . . 10 (𝑛 No → ∅ ≠ 𝑆)
2522, 24eqnetrd 2857 . . . . . . . . 9 (𝑛 No → (𝑛‘( bday 𝑛)) ≠ 𝑆)
2621, 25syl 17 . . . . . . . 8 ((𝐹 No 𝑛𝐹) → (𝑛‘( bday 𝑛)) ≠ 𝑆)
27263adant2 1078 . . . . . . 7 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → (𝑛‘( bday 𝑛)) ≠ 𝑆)
28 fveq2 6148 . . . . . . . . 9 (𝑏 = ( bday 𝑛) → (𝑛𝑏) = (𝑛‘( bday 𝑛)))
2928neeq1d 2849 . . . . . . . 8 (𝑏 = ( bday 𝑛) → ((𝑛𝑏) ≠ 𝑆 ↔ (𝑛‘( bday 𝑛)) ≠ 𝑆))
3029rspcev 3295 . . . . . . 7 ((( bday 𝑛) ∈ suc ( bday 𝐹) ∧ (𝑛‘( bday 𝑛)) ≠ 𝑆) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
3120, 27, 30syl2anc 692 . . . . . 6 ((𝐹 No 𝐹 ∈ V ∧ 𝑛𝐹) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
32313expa 1262 . . . . 5 (((𝐹 No 𝐹 ∈ V) ∧ 𝑛𝐹) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
3332ralrimiva 2960 . . . 4 ((𝐹 No 𝐹 ∈ V) → ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆)
34 rexeq 3128 . . . . . 6 (𝑎 = suc ( bday 𝐹) → (∃𝑏𝑎 (𝑛𝑏) ≠ 𝑆 ↔ ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆))
3534ralbidv 2980 . . . . 5 (𝑎 = suc ( bday 𝐹) → (∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆 ↔ ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆))
3635intminss 4468 . . . 4 ((suc ( bday 𝐹) ∈ On ∧ ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑆) → {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆} ⊆ suc ( bday 𝐹))
377, 33, 36syl2anc 692 . . 3 ((𝐹 No 𝐹 ∈ V) → {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆} ⊆ suc ( bday 𝐹))
385, 37eqsstrd 3618 . 2 ((𝐹 No 𝐹 ∈ V) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))
391, 38sylan2 491 1 ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  {crab 2911  Vcvv 3186   ⊆ wss 3555  ∅c0 3891  {csn 4148  {cpr 4150  ∪ cuni 4402  ∩ cint 4440   × cxp 5072   “ cima 5077  Oncon0 5682  suc csuc 5684   Fn wfn 5842  ‘cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499   No csur 31494   bday cbday 31496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1o 7505  df-2o 7506  df-no 31497  df-bday 31499 This theorem is referenced by:  nobndup  31563  nobnddown  31564
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