Theorem on0eqelt 3119

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 2297	. . 3 |- (/) (_ A
2		0elon 3017	. . . 4 |- (/) e. On
3		onsseleq 2994	. . . 4 |- (((/) e. On /\ A e. On) -> ((/) (_ A <-> ((/) e. A \/ (/) = A)))
4	2, 3	mpan 694	. . 3 |- (A e. On -> ((/) (_ A <-> ((/) e. A \/ (/) = A)))
5	1, 4	mpbii 193	. 2 |- (A e. On -> ((/) e. A \/ (/) = A))
6		eqcom 1474	. . . 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 954   e. wcel 956   (_ wss 2043  (/)c0 2276  Oncon0 2943

This theorem is referenced by:  onxpdisj 3236

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947

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