Theorem infeq5 5727

Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 5733.) The left-hand side provides us with a very short way to express of the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity.

Assertion

Ref	Expression
infeq5	|- (E.x x C. U.x <-> om e. _V)

Proof of Theorem infeq5

Step	Hyp	Ref	Expression
1		df-pss 2607	. . . . 5 |- (x C. U.x <-> (x C_ U.x /\ x =/= U.x))
2		unieq 3185	. . . . . . . . . 10 |- (x = (/) -> U.x = U.(/))
3		uni0 3205	. . . . . . . . . 10 |- U.(/) = (/)
4	2, 3	syl6req 1945	. . . . . . . . 9 |- (x = (/) -> (/) = U.x)
5		eqtr 1904	. . . . . . . . 9 |- ((x = (/) /\ (/) = U.x) -> x = U.x)
6	4, 5	mpdan 768	. . . . . . . 8 |- (x = (/) -> x = U.x)
7	6	necon3i 2042	. . . . . . 7 |- (x =/= U.x -> x =/= (/))
8	7	anim1i 361	. . . . . 6 |- ((x =/= U.x /\ x C_ U.x) -> (x =/= (/) /\ x C_ U.x))
9	8	ancoms 484	. . . . 5 |- ((x C_ U.x /\ x =/= U.x) -> (x =/= (/) /\ x C_ U.x))
10	1, 9	sylbi 216	. . . 4 |- (x C. U.x -> (x =/= (/) /\ x C_ U.x))
11	10	eximi 1387	. . 3 |- (E.x x C. U.x -> E.x(x =/= (/) /\ x C_ U.x))
12		eqid 1884	. . . . 5 |- {<.y, z>. | z = {w e. x | (w i^i x) C_ y}} = {<.y, z>. | z = {w e. x | (w i^i x) C_ y}}
13		eqid 1884	. . . . 5 |- (rec({<.y, z>. | z = {w e. x | (w i^i x) C_ y}}, (/)) |` om) = (rec({<.y, z>. | z = {w e. x | (w i^i x) C_ y}}, (/)) |` om)
14		visset 2295	. . . . 5 |- x e. _V
15	12, 13, 14, 14	inf3lem7 5725	. . . 4 |- ((x =/= (/) /\ x C_ U.x) -> om e. _V)
16	15	19.23aiv 1674	. . 3 |- (E.x(x =/= (/) /\ x C_ U.x) -> om e. _V)
17	11, 16	syl 12	. 2 |- (E.x x C. U.x -> om e. _V)
18		difexg 3458	. . 3 |- (om e. _V -> (om \ {(/)}) e. _V)
19		0ex 3446	. . . . . . 7 |- (/) e. _V
20	19	snid 3069	. . . . . 6 |- (/) e. {(/)}
21		disj4 2922	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> -. (om \ {(/)}) C. om)
22		disj3 2918	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> om = (om \ {(/)}))
23	21, 22	bitr3i 192	. . . . . . . 8 |- (-. (om \ {(/)}) C. om <-> om = (om \ {(/)}))
24		peano1 3971	. . . . . . . . . . 11 |- (/) e. om
25		eleq2 1958	. . . . . . . . . . 11 |- (om = (om \ {(/)}) -> ((/) e. om <-> (/) e. (om \ {(/)})))
26	24, 25	mpbii 210	. . . . . . . . . 10 |- (om = (om \ {(/)}) -> (/) e. (om \ {(/)}))
27		eldif 2609	. . . . . . . . . 10 |- ((/) e. (om \ {(/)}) <-> ((/) e. om /\ -. (/) e. {(/)}))
28	26, 27	sylib 215	. . . . . . . . 9 |- (om = (om \ {(/)}) -> ((/) e. om /\ -. (/) e. {(/)}))
29	28	simprd 352	. . . . . . . 8 |- (om = (om \ {(/)}) -> -. (/) e. {(/)})
30	23, 29	sylbi 216	. . . . . . 7 |- (-. (om \ {(/)}) C. om -> -. (/) e. {(/)})
31	30	con4i 90	. . . . . 6 |- ((/) e. {(/)} -> (om \ {(/)}) C. om)
32	20, 31	ax-mp 7	. . . . 5 |- (om \ {(/)}) C. om
33		unidif0 3476	. . . . . . 7 |- U.(om \ {(/)}) = U.om
34		limom 3967	. . . . . . . 8 |- Lim om
35		limuni 3724	. . . . . . . 8 |- (Lim om -> om = U.om)
36	34, 35	ax-mp 7	. . . . . . 7 |- om = U.om
37	33, 36	eqtr4i 1911	. . . . . 6 |- U.(om \ {(/)}) = om
38	37	psseq2i 2700	. . . . 5 |- ((om \ {(/)}) C. U.(om \ {(/)}) <-> (om \ {(/)}) C. om)
39	32, 38	mpbir 207	. . . 4 |- (om \ {(/)}) C. U.(om \ {(/)})
40		psseq1 2697	. . . . . 6 |- (x = (om \ {(/)}) -> (x C. U.x <-> (om \ {(/)}) C. U.x))
41		unieq 3185	. . . . . . 7 |- (x = (om \ {(/)}) -> U.x = U.(om \ {(/)}))
42	41	psseq2d 2703	. . . . . 6 |- (x = (om \ {(/)}) -> ((om \ {(/)}) C. U.x <-> (om \ {(/)}) C. U.(om \ {(/)})))
43	40, 42	bitrd 587	. . . . 5 |- (x = (om \ {(/)}) -> (x C. U.x <-> (om \ {(/)}) C. U.(om \ {(/)})))
44	43	cla4egv 2365	. . . 4 |- ((om \ {(/)}) e. _V -> ((om \ {(/)}) C. U.(om \ {(/)}) -> E.x x C. U.x))
45	39, 44	mpi 55	. . 3 |- ((om \ {(/)}) e. _V -> E.x x C. U.x)
46	18, 45	syl 12	. 2 |- (om e. _V -> E.x x C. U.x)
47	17, 46	impbii 174	1 |- (E.x x C. U.x <-> om e. _V)

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  {crab 2108  _Vcvv 2292   \ cdif 2590   i^i cin 2592   C_ wss 2593   C. wpss 2594  (/)c0 2875  {csn 3044  U.cuni 3177  {copab 3395  Lim wlim 3658  omcom 3949   |` cres 3988  reccrdg 5139

This theorem is referenced by:  inf5 5735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fv 4014  df-rdg 5140

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