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Theorem nodenselem7 33097
Description: Lemma for nodense 33099. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
Assertion
Ref Expression
nodenselem7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝐶,𝑎

Proof of Theorem nodenselem7
StepHypRef Expression
1 simpll 763 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
2 simplr 765 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
3 sltso 33084 . . . . . . . . 9 <s Or No
4 sonr 5495 . . . . . . . . 9 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
53, 4mpan 686 . . . . . . . 8 (𝐴 No → ¬ 𝐴 <s 𝐴)
6 breq2 5067 . . . . . . . . 9 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
76notbid 319 . . . . . . . 8 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
85, 7syl5ibcom 246 . . . . . . 7 (𝐴 No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
98necon2ad 3036 . . . . . 6 (𝐴 No → (𝐴 <s 𝐵𝐴𝐵))
109imp 407 . . . . 5 ((𝐴 No 𝐴 <s 𝐵) → 𝐴𝐵)
1110ad2ant2rl 745 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴𝐵)
121, 2, 113jca 1122 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 No 𝐵 No 𝐴𝐵))
13 nosepeq 33092 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝐶) = (𝐵𝐶))
1412, 13sylan 580 . 2 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝐶) = (𝐵𝐶))
1514ex 413 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  {crab 3147   cint 4874   class class class wbr 5063   Or wor 5472  Oncon0 6190  cfv 6354   No csur 33050   <s cslt 33051   bday cbday 33052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-1o 8098  df-2o 8099  df-no 33053  df-slt 33054
This theorem is referenced by:  nodense  33099
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