Theorem inf2 8061

Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8077 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)

Hypothesis

Ref	Expression
inf1.1

Assertion

Ref	Expression
inf2

Distinct variable group:   ,,

Proof of Theorem inf2

Step	Hyp	Ref	Expression
1		inf1.1	. . 3
2	1	inf1 8060	. 2
3		dfss2 3492	. . . . 5
4		eluni 4252	. . . . . . 7
5	4	imbi2i 312	. . . . . 6
6	5	albii 1640	. . . . 5
7	3, 6	bitri 249	. . . 4
8	7	anbi2i 694	. . 3
9	8	exbii 1667	. 2
10	2, 9	mpbir 209	1

Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818  =/=wne 2652  C_wss 3475  c0 3784  U.cuni 4249

This theorem is referenced by:  axinf2  8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-uni 4250