Theorem inf1 4607

Description: Variation of Axiom of Infinity (using axinf 4623 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283.

Hypothesis

Ref	Expression
inf1.1	|- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Assertion

Ref	Expression
inf1	|- E.x(x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Distinct variable group:   x,y,z

Proof of Theorem inf1

Step	Hyp	Ref	Expression
1		inf1.1	. 2 |- E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))
2		ne0i 2286	. . . 4 |- (y e. x -> x =/= (/))
3	2	anim1i 334	. . 3 |- ((y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))) -> (x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
4	3	19.22i 1040	. 2 |- (E.x(y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))) -> E.x(x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
5	1, 4	ax-mp 7	1 |- E.x(x =/= (/) /\ A.y(y e. x -> E.z(y e. z /\ z e. x)))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980   =/= wne 1585  (/)c0 2280

This theorem is referenced by:  inf2 4608

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-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-nul 2281

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