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Theorem nodenselem5 33891
Description: Lemma for nodense 33895. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 33890 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem5
StepHypRef Expression
1 simpll 764 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
2 simplr 766 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
3 sltso 33879 . . . . . . . . . 10 <s Or No
4 sonr 5526 . . . . . . . . . 10 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
53, 4mpan 687 . . . . . . . . 9 (𝐴 No → ¬ 𝐴 <s 𝐴)
6 breq2 5078 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
76notbid 318 . . . . . . . . 9 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
85, 7syl5ibcom 244 . . . . . . . 8 (𝐴 No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
98necon2ad 2958 . . . . . . 7 (𝐴 No → (𝐴 <s 𝐵𝐴𝐵))
109adantr 481 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴𝐵))
1110imp 407 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → 𝐴𝐵)
1211adantrl 713 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴𝐵)
13 nosepdm 33887 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
141, 2, 12, 13syl3anc 1370 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
15 simprl 768 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐴) = ( bday 𝐵))
1615uneq2d 4097 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐴)) = (( bday 𝐴) ∪ ( bday 𝐵)))
17 unidm 4086 . . . . 5 (( bday 𝐴) ∪ ( bday 𝐴)) = ( bday 𝐴)
1816, 17eqtr3di 2793 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐵)) = ( bday 𝐴))
19 bdayval 33851 . . . . . 6 (𝐴 No → ( bday 𝐴) = dom 𝐴)
201, 19syl 17 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐴) = dom 𝐴)
21 bdayval 33851 . . . . . 6 (𝐵 No → ( bday 𝐵) = dom 𝐵)
222, 21syl 17 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐵) = dom 𝐵)
2320, 22uneq12d 4098 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐵)) = (dom 𝐴 ∪ dom 𝐵))
2418, 23, 203eqtr3d 2786 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (dom 𝐴 ∪ dom 𝐵) = dom 𝐴)
2514, 24eleqtrd 2841 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
2625, 20eleqtrrd 2842 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  {crab 3068  cun 3885   cint 4879   class class class wbr 5074   Or wor 5502  dom cdm 5589  Oncon0 6266  cfv 6433   No csur 33843   <s cslt 33844   bday cbday 33845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848
This theorem is referenced by:  nodenselem8  33894  nodense  33895
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