Theorem po2ne 5585

Description: Two sets related by a partial order are not equal. (Contributed by AV, 13-Mar-2023.)

Assertion

Ref	Expression
po2ne	⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵)

Proof of Theorem po2ne

Step	Hyp	Ref	Expression
1		breq1 5111	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐵))
2		poirr 5581	. . . . . . . . 9 ⊢ ((𝑅 Po 𝑉 ∧ 𝐵 ∈ 𝑉) → ¬ 𝐵𝑅𝐵)
3	2	adantrl 728	. . . . . . . 8 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ¬ 𝐵𝑅𝐵)
4	3	pm2.21d 122	. . . . . . 7 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐵𝑅𝐵 → 𝐴 ≠ 𝐵))
5	4	ex 417	. . . . . 6 ⊢ (𝑅 Po 𝑉 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵𝑅𝐵 → 𝐴 ≠ 𝐵)))
6	5	com13 89	. . . . 5 ⊢ (𝐵𝑅𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑅 Po 𝑉 → 𝐴 ≠ 𝐵)))
7	1, 6	biimtrdi 256	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑅 Po 𝑉 → 𝐴 ≠ 𝐵))))
8	7	com24 96	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝑉 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝑅𝐵 → 𝐴 ≠ 𝐵))))
9	8	3impd 1365	. 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵))
10		ax-1 6	. 2 ⊢ (𝐴 ≠ 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵))
11	9, 10	pm2.61ine 3039	1 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956   class class class wbr 5108   Po wpo 5567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-po 5569

This theorem is referenced by:  chnpof1  18685  prproropf1olem1  48219