Theorem on0eqelt 3124

Description: An ordinal number either equals zero or contains zero.

Assertion

Ref	Expression
on0eqelt	|- (A e. On -> (A = (/) \/ (/) e. A))

Proof of Theorem on0eqelt

Step	Hyp	Ref	Expression
1		0ss 2301	. . 3 |- (/) (_ A
2		0elon 3022	. . . 4 |- (/) e. On
3		onsseleq 2999	. . . 4 |- (((/) e. On /\ A e. On) -> ((/) (_ A <-> ((/) e. A \/ (/) = A)))
4	2, 3	mpan 695	. . 3 |- (A e. On -> ((/) (_ A <-> ((/) e. A \/ (/) = A)))
5	1, 4	mpbii 193	. 2 |- (A e. On -> ((/) e. A \/ (/) = A))
6		eqcom 1477	. . . 4 |- ((/) = A <-> A = (/))
7	6	orbi2i 255	. . 3 |- (((/) e. A \/ (/) = A) <-> ((/) e. A \/ A = (/)))
8		orcom 246	. . 3 |- (((/) e. A \/ A = (/)) <-> (A = (/) \/ (/) e. A))
9	7, 8	bitr 173	. 2 |- (((/) e. A \/ (/) = A) <-> (A = (/) \/ (/) e. A))
10	5, 9	sylib 198	1 |- (A e. On -> (A = (/) \/ (/) e. A))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958   (_ wss 2047  (/)c0 2280  Oncon0 2948

This theorem is referenced by:  onxpdisj 3241

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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