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Theorem nofulllem2 31102
Description: Lemma for nofull (future) . The full statement of the axiom when 𝐿 is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
Assertion
Ref Expression
nofulllem2 (𝐿 = ∅ → (((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:   𝑦,𝑧   𝑧,𝐿   𝑥,𝐿   𝑦,𝑅,𝑧
Allowed substitution hints:   𝑅(𝑥)   𝐿(𝑦)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem nofulllem2
StepHypRef Expression
1 nobnddown 31100 . . 3 ((𝑅 No 𝑅𝑊) → ∃𝑧 No (∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝑅)))
213ad2ant2 1076 . 2 (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝑅)))
3 rzal 4025 . . . . . 6 (𝐿 = ∅ → ∀𝑥𝐿 𝑥 <s 𝑧)
43biantrurd 528 . . . . 5 (𝐿 = ∅ → (∀𝑦𝑅 𝑧 <s 𝑦 ↔ (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦)))
5 uneq1 3722 . . . . . . . . . . 11 (𝐿 = ∅ → (𝐿𝑅) = (∅ ∪ 𝑅))
6 uncom 3719 . . . . . . . . . . . 12 (∅ ∪ 𝑅) = (𝑅 ∪ ∅)
7 un0 3919 . . . . . . . . . . . 12 (𝑅 ∪ ∅) = 𝑅
86, 7eqtri 2632 . . . . . . . . . . 11 (∅ ∪ 𝑅) = 𝑅
95, 8syl6eq 2660 . . . . . . . . . 10 (𝐿 = ∅ → (𝐿𝑅) = 𝑅)
109imaeq2d 5385 . . . . . . . . 9 (𝐿 = ∅ → ( bday “ (𝐿𝑅)) = ( bday 𝑅))
1110unieqd 4382 . . . . . . . 8 (𝐿 = ∅ → ( bday “ (𝐿𝑅)) = ( bday 𝑅))
12 suceq 5707 . . . . . . . 8 ( ( bday “ (𝐿𝑅)) = ( bday 𝑅) → suc ( bday “ (𝐿𝑅)) = suc ( bday 𝑅))
1311, 12syl 17 . . . . . . 7 (𝐿 = ∅ → suc ( bday “ (𝐿𝑅)) = suc ( bday 𝑅))
1413sseq2d 3596 . . . . . 6 (𝐿 = ∅ → (( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)) ↔ ( bday 𝑧) ⊆ suc ( bday 𝑅)))
1514bicomd 212 . . . . 5 (𝐿 = ∅ → (( bday 𝑧) ⊆ suc ( bday 𝑅) ↔ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅))))
164, 15anbi12d 743 . . . 4 (𝐿 = ∅ → ((∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝑅)) ↔ ((∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦) ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
17 df-3an 1033 . . . 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 277 . . 3 (𝐿 = ∅ → ((∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝑅)) ↔ (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
1918rexbidv 3034 . 2 (𝐿 = ∅ → (∃𝑧 No (∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday 𝑅)) ↔ ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
202, 19syl5ib 233 1 (𝐿 = ∅ → (((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 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cun 3538  wss 3540  c0 3874   cuni 4372   class class class wbr 4583  cima 5041  suc csuc 5642  cfv 5804   No csur 31037   <s cslt 31038   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-slt 31041  df-bday 31042
This theorem is referenced by: (None)
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