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Theorem nodenselem5 27059
Description: Lemma for nodense 27063. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 27058 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 766 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
2 simplr 768 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
3 sltso 27047 . . . . . . . . . 10 <s Or No
4 sonr 5572 . . . . . . . . . 10 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
53, 4mpan 689 . . . . . . . . 9 (𝐴 No → ¬ 𝐴 <s 𝐴)
6 breq2 5113 . . . . . . . . . 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 2955 . . . . . . 7 (𝐴 No → (𝐴 <s 𝐵𝐴𝐵))
109adantr 482 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴𝐵))
1110imp 408 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → 𝐴𝐵)
1211adantrl 715 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴𝐵)
13 nosepdm 27055 . . . 4 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
141, 2, 12, 13syl3anc 1372 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
15 simprl 770 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ( bday 𝐴) = ( bday 𝐵))
1615uneq2d 4127 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐴)) = (( bday 𝐴) ∪ ( bday 𝐵)))
17 unidm 4116 . . . . 5 (( bday 𝐴) ∪ ( bday 𝐴)) = ( bday 𝐴)
1816, 17eqtr3di 2788 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐵)) = ( bday 𝐴))
19 bdayval 27019 . . . . . 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 27019 . . . . . 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 4128 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (( bday 𝐴) ∪ ( bday 𝐵)) = (dom 𝐴 ∪ dom 𝐵))
2418, 23, 203eqtr3d 2781 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (dom 𝐴 ∪ dom 𝐵) = dom 𝐴)
2514, 24eleqtrd 2836 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ dom 𝐴)
2625, 20eleqtrrd 2837 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 397   = wceq 1542  wcel 2107  wne 2940  {crab 3406  cun 3912   cint 4911   class class class wbr 5109   Or wor 5548  dom cdm 5637  Oncon0 6321  cfv 6500   No csur 27011   <s cslt 27012   bday cbday 27013
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-1o 8416  df-2o 8417  df-no 27014  df-slt 27015  df-bday 27016
This theorem is referenced by:  nodenselem8  27062  nodense  27063
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