Theorem onirri 4596

Description: An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
onirri.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onirri	⊢ ¬ 𝐴 ∈ 𝐴

Proof of Theorem onirri

Step	Hyp	Ref	Expression
1		onirri.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onordi 4478	. 2 ⊢ Ord 𝐴
3		ordirr 4595	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2177  Ord word 4414  Oncon0 4415

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-setind 4590

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3170  df-in 3174  df-ss 3181  df-sn 3641  df-uni 3854  df-tr 4148  df-iord 4418  df-on 4420

This theorem is referenced by:  ontri2orexmidim  4625  enpr2d  6922  pm54.43  7310  pw1ne1  7354