Theorem nordeq 4592

Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)

Assertion

Ref	Expression
nordeq	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵)

Proof of Theorem nordeq

Step	Hyp	Ref	Expression
1		ordirr 4590	. . . 4 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		eleq1 2268	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
3	2	notbid 669	. . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐴))
4	1, 3	syl5ibcom 155	. . 3 ⊢ (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
5	4	necon2ad 2433	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐴 ≠ 𝐵))
6	5	imp 124	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176   ≠ wne 2376  Ord word 4409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-sn 3639

This theorem is referenced by:  phplem1  6949