Theorem infeq5 4601

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 4607.) 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 2051	. . . . 5 |- (x (. U.x <-> (x (_ U.x /\ x =/= U.x))
2		unieq 2505	. . . . . . . . . 10 |- (x = (/) -> U.x = U.(/))
3		uni0 2520	. . . . . . . . . 10 |- U.(/) = (/)
4	2, 3	syl6req 1521	. . . . . . . . 9 |- (x = (/) -> (/) = U.x)
5		eqtrt 1489	. . . . . . . . 9 |- ((x = (/) /\ (/) = U.x) -> x = U.x)
6	4, 5	mpdan 703	. . . . . . . 8 |- (x = (/) -> x = U.x)
7	6	necon3i 1602	. . . . . . 7 |- (x =/= U.x -> x =/= (/))
8	7	anim1i 334	. . . . . 6 |- ((x =/= U.x /\ x (_ U.x) -> (x =/= (/) /\ x (_ U.x))
9	8	ancoms 436	. . . . 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 1038	. . 3 |- (E.x x (. U.x -> E.x(x =/= (/) /\ x (_ U.x))
12		eqid 1473	. . . . 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 1473	. . . . 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 1809	. . . . 5 |- x e. V
15	12, 13, 14, 14	inf3lem7 4599	. . . 4 |- ((x =/= (/) /\ x (_ U.x) -> om e. V)
16	15	19.23aiv 1293	. . 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 2717	. . 3 |- (om e. V -> (om \ {(/)}) e. V)
19		0ex 2706	. . . . . . 7 |- (/) e. V
20	19	snid 2431	. . . . . 6 |- (/) e. {(/)}
21		disj4 2313	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> -. (om \ {(/)}) (. om)
22		disj3 2310	. . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> om = (om \ {(/)}))
23	21, 22	bitr3 175	. . . . . . . 8 |- (-. (om \ {(/)}) (. om <-> om = (om \ {(/)}))
24		peano1 3144	. . . . . . . . . . 11 |- (/) e. om
25		eleq2 1532	. . . . . . . . . . 11 |- (om = (om \ {(/)}) -> ((/) e. om <-> (/) e. (om \ {(/)})))
26	24, 25	mpbii 193	. . . . . . . . . 10 |- (om = (om \ {(/)}) -> (/) e. (om \ {(/)}))
27		eldif 2053	. . . . . . . . . 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 2734	. . . . . . 7 |- U.(om \ {(/)}) = U.om
34		limom 3141	. . . . . . . 8 |- Lim om
35		limuni 3024	. . . . . . . 8 |- (Lim om -> om = U.om)
36	34, 35	ax-mp 7	. . . . . . 7 |- om = U.om
37	33, 36	eqtr4 1495	. . . . . 6 |- U.(om \ {(/)}) = om
38	37	psseq2i 2134	. . . . 5 |- ((om \ {(/)}) (. U.(om \ {(/)}) <-> (om \ {(/)}) (. om)
39	32, 38	mpbir 190	. . . 4 |- (om \ {(/)}) (. U.(om \ {(/)})
40		psseq1 2131	. . . . . 6 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.x))
41		unieq 2505	. . . . . . 7 |- (x = (om \ {(/)}) -> U.x = U.(om \ {(/)}))
42	41	psseq2d 2137	. . . . . 6 |- (x = (om \ {(/)}) -> ((om \ {(/)}) (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
43	40, 42	bitrd 527	. . . . 5 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
44	43	cla4egv 1859	. . . 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 954   e. wcel 956  E.wex 978   =/= wne 1582  {crab 1645  Vcvv 1807   \ cdif 2040   i^i cin 2042   (_ wss 2043   (. wpss 2044  (/)c0 2276  {csn 2405  U.cuni 2498  {copab 2661  Lim wlim 2944  omcom 3126   |` cres 3167  reccrdg 3922

This theorem is referenced by:  inf5 4608

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fv 3193  df-rdg 3923

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