Theorem po2ne 5606

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 5151	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐵))
2		poirr 5602	. . . . . . . . 9 ⊢ ((𝑅 Po 𝑉 ∧ 𝐵 ∈ 𝑉) → ¬ 𝐵𝑅𝐵)
3	2	adantrl 715	. . . . . . . 8 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ¬ 𝐵𝑅𝐵)
4	3	pm2.21d 121	. . . . . . 7 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐵𝑅𝐵 → 𝐴 ≠ 𝐵))
5	4	ex 412	. . . . . 6 ⊢ (𝑅 Po 𝑉 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵𝑅𝐵 → 𝐴 ≠ 𝐵)))
6	5	com13 88	. . . . 5 ⊢ (𝐵𝑅𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑅 Po 𝑉 → 𝐴 ≠ 𝐵)))
7	1, 6	biimtrdi 252	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑅 Po 𝑉 → 𝐴 ≠ 𝐵))))
8	7	com24 95	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝑉 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝑅𝐵 → 𝐴 ≠ 𝐵))))
9	8	3impd 1346	. 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 3022	1 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937   class class class wbr 5148   Po wpo 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-po 5590

This theorem is referenced by:  prproropf1olem1  46843