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Theorem nodenselem5 32214
Description: Lemma for nodense 32218. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 32213 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 783 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
2 simplr 785 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
3 sltso 32203 . . . . . . . . . 10 <s Or No
4 sonr 5219 . . . . . . . . . 10 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
53, 4mpan 681 . . . . . . . . 9 (𝐴 No → ¬ 𝐴 <s 𝐴)
6 breq2 4813 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
76notbid 309 . . . . . . . . 9 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
85, 7syl5ibcom 236 . . . . . . . 8 (𝐴 No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
98necon2ad 2952 . . . . . . 7 (𝐴 No → (𝐴 <s 𝐵𝐴𝐵))
109adantr 472 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴𝐵))
1110imp 395 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → 𝐴𝐵)
1211adantrl 707 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴𝐵)
13 nosepdm 32210 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
141, 2, 12, 13syl3anc 1490 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
15 unidm 3918 . . . . 5 (( bday 𝐴) ∪ ( bday 𝐴)) = ( bday 𝐴)
16 simprl 787 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐴) = ( bday 𝐵))
1716uneq2d 3929 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐴)) = (( bday 𝐴) ∪ ( bday 𝐵)))
1815, 17syl5reqr 2814 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐵)) = ( bday 𝐴))
19 bdayval 32177 . . . . . 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 32177 . . . . . 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 3930 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐵)) = (dom 𝐴 ∪ dom 𝐵))
2418, 23, 203eqtr3d 2807 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (dom 𝐴 ∪ dom 𝐵) = dom 𝐴)
2514, 24eleqtrd 2846 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
2625, 20eleqtrrd 2847 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 384   = wceq 1652  wcel 2155  wne 2937  {crab 3059  cun 3730   cint 4633   class class class wbr 4809   Or wor 5197  dom cdm 5277  Oncon0 5908  cfv 6068   No csur 32169   <s cslt 32170   bday cbday 32171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-1o 7764  df-2o 7765  df-no 32172  df-slt 32173  df-bday 32174
This theorem is referenced by:  nodenselem8  32217  nodense  32218
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