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Theorem nodenselem4 33340
 Description: Lemma for nodense 33345. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem4 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem4
StepHypRef Expression
1 simpll 766 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → 𝐴 No )
2 simplr 768 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → 𝐵 No )
3 sltso 33330 . . . . . . 7 <s Or No
4 sonr 5461 . . . . . . 7 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
53, 4mpan 689 . . . . . 6 (𝐴 No → ¬ 𝐴 <s 𝐴)
65adantr 484 . . . . 5 ((𝐴 No 𝐵 No ) → ¬ 𝐴 <s 𝐴)
7 breq2 5035 . . . . . 6 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
87notbid 321 . . . . 5 (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
96, 8syl5ibcom 248 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
109necon2ad 3002 . . 3 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴𝐵))
1110imp 410 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → 𝐴𝐵)
12 nosepon 33321 . 2 ((𝐴 No 𝐵 No 𝐴𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
131, 2, 11, 12syl3anc 1368 1 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  ∩ cint 4839   class class class wbr 5031   Or wor 5438  Oncon0 6162  ‘cfv 6327   No csur 33296
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