Theorem inf1 8519

Description: Variation of Axiom of Infinity (using zfinf 8536 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)

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 ) ) )

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 3921	. . 3 |- ( y e. x -> x =/= (/) )
3	2	anim1i 592	. 2 |- ( ( 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	1, 3	eximii 1764	1 |- E. x ( x =/= (/) /\ A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   =/= wne 2794   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  inf2  8520