Theorem nordeq 6379

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 6378	. . . 4 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		eleq1 2851	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
3	2	notbid 321	. . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐴))
4	1, 3	syl5ibcom 248	. . 3 ⊢ (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴))
5	4	necon2ad 2973	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐴 ≠ 𝐵))
6	5	imp 411	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Ord word 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-eprel 5561  df-fr 5614  df-we 5616  df-ord 6363

This theorem is referenced by:  php  9187  nogt01o  27860  ordtop  36967  limsucncmpi  36976