Theorem infeq5 4766

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 4772.) 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 2107	. . . . 5 |- (x (. U.x <-> (x (_ U.x /\ x =/= U.x))
2		unieq 2576	. . . . . . . . . 10 |- (x = (/) -> U.x = U.(/))
3		uni0 2592	. . . . . . . . . 10 |- U.(/) = (/)
4	2, 3	syl6req 1567	. . . . . . . . 9 |- (x = (/) -> (/) = U.x)
5		eqtr 1535	. . . . . . . . 9 |- ((x = (/) /\ (/) = U.x) -> x = U.x)
6	4, 5	mpdan 708	. . . . . . . 8 |- (x = (/) -> x = U.x)
7	6	necon3i 1648	. . . . . . 7 |- (x =/= U.x -> x =/= (/))
8	7	anim1i 332	. . . . . 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 197	. . . 4 |- (x (. U.x -> (x =/= (/) /\ x (_ U.x))
11	10	19.22i 1076	. . 3 |- (E.x x (. U.x -> E.x(x =/= (/) /\ x (_ U.x))
12		eqid 1518	. . . . 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 1518	. . . . 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 1859	. . . . 5 |- x e. V
15	12, 13, 14, 14	inf3lem7 4764	. . . 4 |- ((x =/= (/) /\ x (_ U.x) -> om e. V)
16	15	19.23aiv 1333	. . 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 2796	. . 3 |- (om e. V -> (om \ {(/)}) e. V)
19		0ex 2785	. . . . . . 7 |- (/) e. V
20	19	snid 2496	. . . . . 6 |- (/) e. {(/)}
21		disj4 2370	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> -. (om \ {(/)}) (. om)
22		disj3 2367	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> om = (om \ {(/)}))
23	21, 22	bitr3i 173	. . . . . . . 8 |- (-. (om \ {(/)}) (. om <-> om = (om \ {(/)}))
24		peano1 3237	. . . . . . . . . . 11 |- (/) e. om
25		eleq2 1578	. . . . . . . . . . 11 |- (om = (om \ {(/)}) -> ((/) e. om <-> (/) e. (om \ {(/)})))
26	24, 25	mpbii 191	. . . . . . . . . 10 |- (om = (om \ {(/)}) -> (/) e. (om \ {(/)}))
27		eldif 2109	. . . . . . . . . 10 |- ((/) e. (om \ {(/)}) <-> ((/) e. om /\ -. (/) e. {(/)}))
28	26, 27	sylib 196	. . . . . . . . 9 |- (om = (om \ {(/)}) -> ((/) e. om /\ -. (/) e. {(/)}))
29	28	pm3.27d 323	. . . . . . . 8 |- (om = (om \ {(/)}) -> -. (/) e. {(/)})
30	23, 29	sylbi 197	. . . . . . 7 |- (-. (om \ {(/)}) (. om -> -. (/) e. {(/)})
31	30	con4i 74	. . . . . 6 |- ((/) e. {(/)} -> (om \ {(/)}) (. om)
32	20, 31	ax-mp 7	. . . . 5 |- (om \ {(/)}) (. om
33		unidif0 2813	. . . . . . 7 |- U.(om \ {(/)}) = U.om
34		limom 3233	. . . . . . . 8 |- Lim om
35		limuni 3033	. . . . . . . 8 |- (Lim om -> om = U.om)
36	34, 35	ax-mp 7	. . . . . . 7 |- om = U.om
37	33, 36	eqtr4i 1541	. . . . . 6 |- U.(om \ {(/)}) = om
38	37	psseq2i 2190	. . . . 5 |- ((om \ {(/)}) (. U.(om \ {(/)}) <-> (om \ {(/)}) (. om)
39	32, 38	mpbir 188	. . . 4 |- (om \ {(/)}) (. U.(om \ {(/)})
40		psseq1 2187	. . . . . 6 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.x))
41		unieq 2576	. . . . . . 7 |- (x = (om \ {(/)}) -> U.x = U.(om \ {(/)}))
42	41	psseq2d 2193	. . . . . 6 |- (x = (om \ {(/)}) -> ((om \ {(/)}) (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
43	40, 42	bitrd 531	. . . . 5 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
44	43	cla4egv 1909	. . . 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	impbii 155	1 |- (E.x x (. U.x <-> om e. V)

Syntax hints:  -. wn 2   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  E.wex 1016   =/= wne 1628  {crab 1694  Vcvv 1857   \ cdif 2096   i^i cin 2098   (_ wss 2099   (. wpss 2100  (/)c0 2332  {csn 2467  U.cuni 2569  {copab 2740  Lim wlim 2976  omcom 3218   |` cres 3253  reccrdg 4232

This theorem is referenced by:  inf5 4774

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-reg 4736

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fv 3279  df-rdg 4233

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