Theorem nordeq 4636

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 4634	. . . 4 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		eleq1 2292	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
3	2	notbid 671	. . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐴))
4	1, 3	syl5ibcom 155	. . 3 ⊢ (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
5	4	necon2ad 2457	. 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 1395   ∈ wcel 2200   ≠ wne 2400  Ord word 4453

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-sn 3672

This theorem is referenced by:  phplem1  7013