Theorem po2ne 5482

Description: Two classes which are in a partial order relation 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 5062	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐵))
2		poirr 5478	. . . . . . . . 9 ⊢ ((𝑅 Po 𝑉 ∧ 𝐵 ∈ 𝑉) → ¬ 𝐵𝑅𝐵)
3	2	adantrl 714	. . . . . . . 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 415	. . . . . 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	syl6bi 255	. . . 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 1343	. 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 3099	1 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113   ≠ wne 3015   class class class wbr 5059   Po wpo 5465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-po 5467

This theorem is referenced by:  prproropf1olem1  43735