Theorem inf5 3472

Description: The statement "there exists a set that is the proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 3475.) This provides us with a very short way to express of the Axiom of Infinity using only simple defined symbols. The proof does not depend on the Axiom of Infinity.

Assertion

Ref	Expression
inf5	|- (E.x x (. U.x <-> om e. V)

Proof of Theorem inf5

Step	Hyp	Ref	Expression
1		dfpss2 1557	. . . . . . 7 |- (x (. U.x <-> (x (_ U.x /\ -. x = U.x))
2	1	pm3.27bd 263	. . . . . 6 |- (x (. U.x -> -. x = U.x)
3		unieq 1927	. . . . . . . 8 |- (x = (/) -> U.x = U.(/))
4		uni0 1938	. . . . . . . 8 |- U.(/) = (/)
5	3, 4	syl6req 1141	. . . . . . 7 |- (x = (/) -> (/) = U.x)
6		cleqtr 1118	. . . . . . 7 |- ((x = (/) /\ (/) = U.x) -> x = U.x)
7	5, 6	mpdan 527	. . . . . 6 |- (x = (/) -> x = U.x)
8	2, 7	nsyl 102	. . . . 5 |- (x (. U.x -> -. x = (/))
9		pssss 1567	. . . . 5 |- (x (. U.x -> x (_ U.x)
10	8, 9	jca 236	. . . 4 |- (x (. U.x -> (-. x = (/) /\ x (_ U.x))
11	10	19.22i 723	. . 3 |- (E.x x (. U.x -> E.x(-. x = (/) /\ x (_ U.x))
12		cleqid 1102	. . . . 5 |- {<.y, z>. | z = {w e. x | (w i^i x) (_ y}} = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
13		cleqid 1102	. . . . 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 1350	. . . . 5 |- x e. V
15	12, 13, 14, 14	inf3lem7 3470	. . . 4 |- ((-. x = (/) /\ x (_ U.x) -> om e. V)
16	15	19.23aiv 952	. . 3 |- (E.x(-. x = (/) /\ x (_ U.x) -> om e. V)
17	11, 16	syl 12	. 2 |- (E.x x (. U.x -> om e. V)
18		difexg 1703	. . 3 |- (om e. V -> (om \ {(/)}) e. V)
19		0ex 1745	. . . . . . 7 |- (/) e. V
20	19	snid 1830	. . . . . 6 |- (/) e. {(/)}
21		disj4 1737	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> -. (om \ {(/)}) (. om)
22		disj3 1736	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> om = (om \ {(/)}))
23	21, 22	bitr3 153	. . . . . . . 8 |- (-. (om \ {(/)}) (. om <-> om = (om \ {(/)}))
24		peano1 2390	. . . . . . . . . . 11 |- (/) e. om
25		eleq2 1150	. . . . . . . . . . 11 |- (om = (om \ {(/)}) -> ((/) e. om <-> (/) e. (om \ {(/)})))
26	24, 25	mpbii 168	. . . . . . . . . 10 |- (om = (om \ {(/)}) -> (/) e. (om \ {(/)}))
27		eldif 1496	. . . . . . . . . 10 |- ((/) e. (om \ {(/)}) <-> ((/) e. om /\ -. (/) e. {(/)}))
28	26, 27	sylib 173	. . . . . . . . 9 |- (om = (om \ {(/)}) -> ((/) e. om /\ -. (/) e. {(/)}))
29	28	pm3.27d 262	. . . . . . . 8 |- (om = (om \ {(/)}) -> -. (/) e. {(/)})
30	23, 29	sylbi 174	. . . . . . 7 |- (-. (om \ {(/)}) (. om -> -. (/) e. {(/)})
31	30	a3i 69	. . . . . 6 |- ((/) e. {(/)} -> (om \ {(/)}) (. om)
32	20, 31	ax-mp 6	. . . . 5 |- (om \ {(/)}) (. om
33		unidif0 1944	. . . . . . 7 |- U.(om \ {(/)}) = U.om
34		limom 2387	. . . . . . . 8 |- Lim om
35		limuni 2284	. . . . . . . 8 |- (Lim om -> om = U.om)
36	34, 35	ax-mp 6	. . . . . . 7 |- om = U.om
37	33, 36	eqtr4 1122	. . . . . 6 |- U.(om \ {(/)}) = om
38	37	psseq2i 1562	. . . . 5 |- ((om \ {(/)}) (. U.(om \ {(/)}) <-> (om \ {(/)}) (. om)
39	32, 38	mpbir 165	. . . 4 |- (om \ {(/)}) (. U.(om \ {(/)})
40		psseq1 1559	. . . . . 6 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.x))
41		unieq 1927	. . . . . . 7 |- (x = (om \ {(/)}) -> U.x = U.(om \ {(/)}))
42	41	psseq2d 1565	. . . . . 6 |- (x = (om \ {(/)}) -> ((om \ {(/)}) (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
43	40, 42	bitrd 406	. . . . 5 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
44	43	cla4egv 1397	. . . 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 12	. 2 |- (om e. V -> E.x x (. U.x)
47	17, 46	impbi 139	1 |- (E.x x (. U.x <-> om e. V)

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348   \ cdif 1484   i^i cin 1486   (_ wss 1487   (. wpss 1488  (/)c0 1707  {csn 1808  U.cuni 1919  {copab 2055  Lim wlim 2200  omcom 2372   |` cres 2412  reccrdg 2969

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970

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