Theorem onnmin 3005

Description: No member of a set of ordinal numbers belongs to its minimum.

Assertion

Ref	Expression
onnmin	|- ((A (_ On /\ B e. A) -> -. B e. |^|A)

Proof of Theorem onnmin

Step	Hyp	Ref	Expression
1		intss1 2538	. . 3 |- (B e. A -> |^|A (_ B)
2	1	adantl 388	. 2 |- ((A (_ On /\ B e. A) -> |^|A (_ B)
3		ontri1 2971	. . 3 |- ((|^|A e. On /\ B e. On) -> (|^|A (_ B <-> -. B e. |^|A))
4		oninton 3002	. . . 4 |- ((A (_ On /\ A =/= (/)) -> |^|A e. On)
5		ne0i 2276	. . . 4 |- (B e. A -> A =/= (/))
6	4, 5	sylan2 451	. . 3 |- ((A (_ On /\ B e. A) -> |^|A e. On)
7		ssel2 2054	. . 3 |- ((A (_ On /\ B e. A) -> B e. On)
8	3, 6, 7	sylanc 471	. 2 |- ((A (_ On /\ B e. A) -> (|^|A (_ B <-> -. B e. |^|A))
9	2, 8	mpbid 195	1 |- ((A (_ On /\ B e. A) -> -. B e. |^|A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955   =/= wne 1577   (_ wss 2037  (/)c0 2270  |^|cint 2523  Oncon0 2938

This theorem is referenced by:  onnminsb 3006  oneqmin 3008  onminex 3010  onmindif2 3051  cardmin 4832

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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