Theorem wess 2942

Description: Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31.

Assertion

Ref	Expression
wess	|- (A (_ B -> (R We B -> R We A))

Proof of Theorem wess

Step	Hyp	Ref	Expression
1		frss 2927	. . 3 |- (A (_ B -> (R Fr B -> R Fr A))
2		soss 2858	. . 3 |- (A (_ B -> (R Or B -> R Or A))
3	1, 2	anim12d 560	. 2 |- (A (_ B -> ((R Fr B /\ R Or B) -> (R Fr A /\ R Or A)))
4		df-we 2940	. 2 |- (R We B <-> (R Fr B /\ R Or B))
5		df-we 2940	. 2 |- (R We A <-> (R Fr A /\ R Or A))
6	3, 4, 5	3imtr4g 555	1 |- (A (_ B -> (R We B -> R We A))

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2050   Or wor 2845   Fr wfr 2921   We wwe 2922

This theorem is referenced by:  wefrc 2949  wereu 2951  trssord 2971  ordelord 2976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-po 2846  df-so 2856  df-fr 2923  df-we 2940

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