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Theorem nodenselem7 32138
 Description: Lemma for nodense 32140. 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 807 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 No )
2 simplr 809 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 No )
3 sltso 32125 . . . . . . . . 9 <s Or No
4 sonr 5200 . . . . . . . . 9 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
53, 4mpan 708 . . . . . . . 8 (𝐴 No → ¬ 𝐴 <s 𝐴)
6 breq2 4800 . . . . . . . . 9 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
76notbid 307 . . . . . . . 8 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
85, 7syl5ibcom 235 . . . . . . 7 (𝐴 No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
98necon2ad 2939 . . . . . 6 (𝐴 No → (𝐴 <s 𝐵𝐴𝐵))
109imp 444 . . . . 5 ((𝐴 No 𝐴 <s 𝐵) → 𝐴𝐵)
1110ad2ant2rl 802 . . . 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 32133 . . 3 (((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝐶) = (𝐵𝐶))
1412, 13sylan 489 . 2 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐴𝐶) = (𝐵𝐶))
1514ex 449 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 383   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  {crab 3046  ∩ cint 4619   class class class wbr 4796   Or wor 5178  Oncon0 5876  ‘cfv 6041   No csur 32091
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