Theorem nordeq 2218

Description: A member of an ordinal class is not equal to it.

Assertion

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

Proof of Theorem nordeq

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . 6 ⊢ (A = B → (A ∈ A ↔ B ∈ A))
2	1	negbid 463	. . . . 5 ⊢ (A = B → (¬ A ∈ A ↔ ¬ B ∈ A))
3		ordeirr 2217	. . . . 5 ⊢ (Ord A → ¬ A ∈ A)
4	2, 3	syl5bi 183	. . . 4 ⊢ (A = B → (Ord A → ¬ B ∈ A))
5	4	com12 13	. . 3 ⊢ (Ord A → (A = B → ¬ B ∈ A))
6	5	con2d 83	. 2 ⊢ (Ord A → (B ∈ A → ¬ A = B))
7	6	imp 277	1 ⊢ ((Ord A ∧ B ∈ A) → ¬ A = B)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Ord word 2198

This theorem is referenced by:  phplem2 3404  php 3409

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169  df-we 2186  df-ord 2202

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