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Theorem nodenselem3 31082
 Description: Lemma for nodense 31088. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem3 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem3
StepHypRef Expression
1 bdayval 31045 . . . 4 (𝐵 No → ( bday 𝐵) = dom 𝐵)
21adantl 481 . . 3 ((𝐴 No 𝐵 No ) → ( bday 𝐵) = dom 𝐵)
32eleq2d 2673 . 2 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) ↔ ( bday 𝐴) ∈ dom 𝐵))
4 bdayelon 31079 . . . 4 ( bday 𝐴) ∈ On
5 nosgnn0 31055 . . . . . . . . 9 ¬ ∅ ∈ {1𝑜, 2𝑜}
6 norn 31048 . . . . . . . . . . . 12 (𝐵 No → ran 𝐵 ⊆ {1𝑜, 2𝑜})
76adantr 480 . . . . . . . . . . 11 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → ran 𝐵 ⊆ {1𝑜, 2𝑜})
8 nofun 31046 . . . . . . . . . . . 12 (𝐵 No → Fun 𝐵)
9 fvelrn 6260 . . . . . . . . . . . 12 ((Fun 𝐵 ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ∈ ran 𝐵)
108, 9sylan 487 . . . . . . . . . . 11 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ∈ ran 𝐵)
117, 10sseldd 3569 . . . . . . . . . 10 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ∈ {1𝑜, 2𝑜})
12 eleq1 2676 . . . . . . . . . 10 ((𝐵‘( bday 𝐴)) = ∅ → ((𝐵‘( bday 𝐴)) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
1311, 12syl5ibcom 234 . . . . . . . . 9 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → ((𝐵‘( bday 𝐴)) = ∅ → ∅ ∈ {1𝑜, 2𝑜}))
145, 13mtoi 189 . . . . . . . 8 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → ¬ (𝐵‘( bday 𝐴)) = ∅)
1514neqned 2789 . . . . . . 7 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ≠ ∅)
1615adantll 746 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ≠ ∅)
17 fvnobday 31081 . . . . . . 7 (𝐴 No → (𝐴‘( bday 𝐴)) = ∅)
1817ad2antrr 758 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐴‘( bday 𝐴)) = ∅)
1916, 18neeqtrrd 2856 . . . . 5 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ≠ (𝐴‘( bday 𝐴)))
2019necomd 2837 . . . 4 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐴‘( bday 𝐴)) ≠ (𝐵‘( bday 𝐴)))
21 fveq2 6103 . . . . . 6 (𝑎 = ( bday 𝐴) → (𝐴𝑎) = (𝐴‘( bday 𝐴)))
22 fveq2 6103 . . . . . 6 (𝑎 = ( bday 𝐴) → (𝐵𝑎) = (𝐵‘( bday 𝐴)))
2321, 22neeq12d 2843 . . . . 5 (𝑎 = ( bday 𝐴) → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴‘( bday 𝐴)) ≠ (𝐵‘( bday 𝐴))))
2423rspcev 3282 . . . 4 ((( bday 𝐴) ∈ On ∧ (𝐴‘( bday 𝐴)) ≠ (𝐵‘( bday 𝐴))) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
254, 20, 24sylancr 694 . . 3 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
2625ex 449 . 2 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ dom 𝐵 → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
273, 26sylbid 229 1 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  {cpr 4127  dom cdm 5038  ran crn 5039  Oncon0 5640  Fun wfun 5798  ‘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-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:  nodenselem4  31083
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