Theorem wess 2936

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 2921	. . 3 |- (A (_ B -> (R Fr B -> R Fr A))
2		soss 2852	. . 3 |- (A (_ B -> (R Or B -> R Or A))
3	1, 2	anim12d 558	. 2 |- (A (_ B -> ((R Fr B /\ R Or B) -> (R Fr A /\ R Or A)))
4		df-we 2934	. 2 |- (R We B <-> (R Fr B /\ R Or B))
5		df-we 2934	. 2 |- (R We A <-> (R Fr A /\ R Or A))
6	3, 4, 5	3imtr4g 553	1 |- (A (_ B -> (R We B -> R We A))

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2047   Or wor 2839   Fr wfr 2915   We wwe 2916

This theorem is referenced by:  wefrc 2943  wereu 2945  trssord 2965  ordelord 2970

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  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-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-po 2840  df-so 2850  df-fr 2917  df-we 2934

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