Theorem infeq5 4630

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 4636.) 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 (. U.x <-> om e. V)

Proof of Theorem infeq5

Step	Hyp	Ref	Expression
1		df-pss 2058	. . . . 5 |- (x (. U.x <-> (x (_ U.x /\ x =/= U.x))
2		unieq 2514	. . . . . . . . . 10 |- (x = (/) -> U.x = U.(/))
3		uni0 2529	. . . . . . . . . 10 |- U.(/) = (/)
4	2, 3	syl6req 1527	. . . . . . . . 9 |- (x = (/) -> (/) = U.x)
5		eqtrt 1495	. . . . . . . . 9 |- ((x = (/) /\ (/) = U.x) -> x = U.x)
6	4, 5	mpdan 706	. . . . . . . 8 |- (x = (/) -> x = U.x)
7	6	necon3i 1608	. . . . . . 7 |- (x =/= U.x -> x =/= (/))
8	7	anim1i 334	. . . . . 6 |- ((x =/= U.x /\ x (_ U.x) -> (x =/= (/) /\ x (_ U.x))
9	8	ancoms 438	. . . . 5 |- ((x (_ U.x /\ x =/= U.x) -> (x =/= (/) /\ x (_ U.x))
10	1, 9	sylbi 199	. . . 4 |- (x (. U.x -> (x =/= (/) /\ x (_ U.x))
11	10	19.22i 1042	. . 3 |- (E.x x (. U.x -> E.x(x =/= (/) /\ x (_ U.x))
12		eqid 1478	. . . . 5 |- {<.y, z>. | z = {w e. x | (w i^i x) (_ y}} = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
13		eqid 1478	. . . . 5 |- (rec({<.y, z>. | z = {w e. x | (w i^i x) (_ y}}, (/)) |` om) = (rec({<.y, z>. | z = {w e. x | (w i^i x) (_ y}}, (/)) |` om)
14		visset 1816	. . . . 5 |- x e. V
15	12, 13, 14, 14	inf3lem7 4628	. . . 4 |- ((x =/= (/) /\ x (_ U.x) -> om e. V)
16	15	19.23aiv 1297	. . 3 |- (E.x(x =/= (/) /\ x (_ U.x) -> om e. V)
17	11, 16	syl 10	. 2 |- (E.x x (. U.x -> om e. V)
18		difexg 2727	. . 3 |- (om e. V -> (om \ {(/)}) e. V)
19		0ex 2716	. . . . . . 7 |- (/) e. V
20	19	snid 2439	. . . . . 6 |- (/) e. {(/)}
21		disj4 2321	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> -. (om \ {(/)}) (. om)
22		disj3 2318	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> om = (om \ {(/)}))
23	21, 22	bitr3 175	. . . . . . . 8 |- (-. (om \ {(/)}) (. om <-> om = (om \ {(/)}))
24		peano1 3155	. . . . . . . . . . 11 |- (/) e. om
25		eleq2 1538	. . . . . . . . . . 11 |- (om = (om \ {(/)}) -> ((/) e. om <-> (/) e. (om \ {(/)})))
26	24, 25	mpbii 193	. . . . . . . . . 10 |- (om = (om \ {(/)}) -> (/) e. (om \ {(/)}))
27		eldif 2060	. . . . . . . . . 10 |- ((/) e. (om \ {(/)}) <-> ((/) e. om /\ -. (/) e. {(/)}))
28	26, 27	sylib 198	. . . . . . . . 9 |- (om = (om \ {(/)}) -> ((/) e. om /\ -. (/) e. {(/)}))
29	28	pm3.27d 325	. . . . . . . 8 |- (om = (om \ {(/)}) -> -. (/) e. {(/)})
30	23, 29	sylbi 199	. . . . . . 7 |- (-. (om \ {(/)}) (. om -> -. (/) e. {(/)})
31	30	a3i 74	. . . . . 6 |- ((/) e. {(/)} -> (om \ {(/)}) (. om)
32	20, 31	ax-mp 7	. . . . 5 |- (om \ {(/)}) (. om
33		unidif0 2744	. . . . . . 7 |- U.(om \ {(/)}) = U.om
34		limom 3152	. . . . . . . 8 |- Lim om
35		limuni 3035	. . . . . . . 8 |- (Lim om -> om = U.om)
36	34, 35	ax-mp 7	. . . . . . 7 |- om = U.om
37	33, 36	eqtr4 1501	. . . . . 6 |- U.(om \ {(/)}) = om
38	37	psseq2i 2141	. . . . 5 |- ((om \ {(/)}) (. U.(om \ {(/)}) <-> (om \ {(/)}) (. om)
39	32, 38	mpbir 190	. . . 4 |- (om \ {(/)}) (. U.(om \ {(/)})
40		psseq1 2138	. . . . . 6 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.x))
41		unieq 2514	. . . . . . 7 |- (x = (om \ {(/)}) -> U.x = U.(om \ {(/)}))
42	41	psseq2d 2144	. . . . . 6 |- (x = (om \ {(/)}) -> ((om \ {(/)}) (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
43	40, 42	bitrd 530	. . . . 5 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
44	43	cla4egv 1866	. . . 4 |- ((om \ {(/)}) e. V -> ((om \ {(/)}) (. U.(om \ {(/)}) -> E.x x (. U.x))
45	39, 44	mpi 44	. . 3 |- ((om \ {(/)}) e. V -> E.x x (. U.x)
46	18, 45	syl 10	. 2 |- (om e. V -> E.x x (. U.x)
47	17, 46	impbi 157	1 |- (E.x x (. U.x <-> om e. V)

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  {crab 1651  Vcvv 1814   \ cdif 2047   i^i cin 2049   (_ wss 2050   (. wpss 2051  (/)c0 2283  {csn 2413  U.cuni 2507  {copab 2671  Lim wlim 2955  omcom 3137   |` cres 3178  reccrdg 3937

This theorem is referenced by:  inf5 4637

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fv 3204  df-rdg 3938

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