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Theorem nofulllem1 30936
Description: Lemma for nofull (future) . The full statement of the axiom when 𝑅 is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
Assertion
Ref Expression
nofulllem1 (𝑅 = ∅ → (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
Distinct variable groups:   𝑥,𝑧,𝐿   𝑦,𝑅   𝑧,𝑅
Allowed substitution hints:   𝑅(𝑥)   𝐿(𝑦)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem nofulllem1
StepHypRef Expression
1 nobndup 30934 . . 3 ((𝐿 No 𝐿𝑉) → ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿)))
213ad2ant1 1074 . 2 (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿)))
3 raleq 3019 . . . . . . 7 (𝑅 = ∅ → (∀𝑦𝑅 𝑧 <s 𝑦 ↔ ∀𝑦 ∈ ∅ 𝑧 <s 𝑦))
4 uneq2 3627 . . . . . . . . . . . 12 (𝑅 = ∅ → (𝐿𝑅) = (𝐿 ∪ ∅))
5 un0 3822 . . . . . . . . . . . 12 (𝐿 ∪ ∅) = 𝐿
64, 5syl6eq 2564 . . . . . . . . . . 11 (𝑅 = ∅ → (𝐿𝑅) = 𝐿)
76imaeq2d 5276 . . . . . . . . . 10 (𝑅 = ∅ → ( bday “ (𝐿𝑅)) = ( bday 𝐿))
87unieqd 4280 . . . . . . . . 9 (𝑅 = ∅ → ( bday “ (𝐿𝑅)) = ( bday 𝐿))
9 suceq 5595 . . . . . . . . 9 ( ( bday “ (𝐿𝑅)) = ( bday 𝐿) → suc ( bday “ (𝐿𝑅)) = suc ( bday 𝐿))
108, 9syl 17 . . . . . . . 8 (𝑅 = ∅ → suc ( bday “ (𝐿𝑅)) = suc ( bday 𝐿))
1110sseq2d 3500 . . . . . . 7 (𝑅 = ∅ → (( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)) ↔ ( bday 𝑧) ⊆ suc ( bday 𝐿)))
123, 11anbi12d 742 . . . . . 6 (𝑅 = ∅ → ((∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅))) ↔ (∀𝑦 ∈ ∅ 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿))))
13 ral0 3931 . . . . . . 7 𝑦 ∈ ∅ 𝑧 <s 𝑦
1413biantrur 525 . . . . . 6 (( bday 𝑧) ⊆ suc ( bday 𝐿) ↔ (∀𝑦 ∈ ∅ 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿)))
1512, 14syl6rbbr 277 . . . . 5 (𝑅 = ∅ → (( bday 𝑧) ⊆ suc ( bday 𝐿) ↔ (∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
1615anbi2d 735 . . . 4 (𝑅 = ∅ → ((∀𝑥𝐿 𝑥 <s 𝑧 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿)) ↔ (∀𝑥𝐿 𝑥 <s 𝑧 ∧ (∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅))))))
17 3anass 1034 . . . 4 ((∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅))) ↔ (∀𝑥𝐿 𝑥 <s 𝑧 ∧ (∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
1816, 17syl6bbr 276 . . 3 (𝑅 = ∅ → ((∀𝑥𝐿 𝑥 <s 𝑧 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿)) ↔ (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
1918rexbidv 2938 . 2 (𝑅 = ∅ → (∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ( bday 𝑧) ⊆ suc ( bday 𝐿)) ↔ ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
202, 19syl5ib 232 1 (𝑅 = ∅ → (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800  wrex 2801  cun 3442  wss 3444  c0 3777   cuni 4270   class class class wbr 4481  cima 4935  suc csuc 5532  cfv 5689   No csur 30872   <s cslt 30873   bday cbday 30874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-1o 7323  df-2o 7324  df-no 30875  df-slt 30876  df-bday 30877
This theorem is referenced by: (None)
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