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	⊢ (∃x x ⊂ ∪x ↔ ω ∈ V)

Proof of Theorem inf5

Step	Hyp	Ref	Expression
1		dfpss2 1557	. . . . . . 7 ⊢ (x ⊂ ∪x ↔ (x ⊆ ∪x ∧ ¬ x = ∪x))
2	1	pm3.27bd 263	. . . . . 6 ⊢ (x ⊂ ∪x → ¬ x = ∪x)
3		unieq 1927	. . . . . . . 8 ⊢ (x = ∅ → ∪x = ∪∅)
4		uni0 1938	. . . . . . . 8 ⊢ ∪∅ = ∅
5	3, 4	syl6req 1141	. . . . . . 7 ⊢ (x = ∅ → ∅ = ∪x)
6		cleqtr 1118	. . . . . . 7 ⊢ ((x = ∅ ∧ ∅ = ∪x) → x = ∪x)
7	5, 6	mpdan 527	. . . . . 6 ⊢ (x = ∅ → x = ∪x)
8	2, 7	nsyl 102	. . . . 5 ⊢ (x ⊂ ∪x → ¬ x = ∅)
9		pssss 1567	. . . . 5 ⊢ (x ⊂ ∪x → x ⊆ ∪x)
10	8, 9	jca 236	. . . 4 ⊢ (x ⊂ ∪x → (¬ x = ∅ ∧ x ⊆ ∪x))
11	10	19.22i 723	. . 3 ⊢ (∃x x ⊂ ∪x → ∃x(¬ x = ∅ ∧ x ⊆ ∪x))
12		cleqid 1102	. . . . 5 ⊢ {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}} = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
13		cleqid 1102	. . . . 5 ⊢ (rec({⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}, ∅) ↾ ω) = (rec({⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}, ∅) ↾ ω)
14		visset 1350	. . . . 5 ⊢ x ∈ V
15	12, 13, 14, 14	inf3lem7 3470	. . . 4 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → ω ∈ V)
16	15	19.23aiv 952	. . 3 ⊢ (∃x(¬ x = ∅ ∧ x ⊆ ∪x) → ω ∈ V)
17	11, 16	syl 12	. 2 ⊢ (∃x x ⊂ ∪x → ω ∈ V)
18		difexg 1703	. . 3 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V)
19		0ex 1745	. . . . . . 7 ⊢ ∅ ∈ V
20	19	snid 1830	. . . . . 6 ⊢ ∅ ∈ {∅}
21		disj4 1737	. . . . . . . . 9 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊂ ω)
22		disj3 1736	. . . . . . . . 9 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅}))
23	21, 22	bitr3 153	. . . . . . . 8 ⊢ (¬ (ω ∖ {∅}) ⊂ ω ↔ ω = (ω ∖ {∅}))
24		peano1 2390	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
25		eleq2 1150	. . . . . . . . . . 11 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅})))
26	24, 25	mpbii 168	. . . . . . . . . 10 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅}))
27		eldif 1496	. . . . . . . . . 10 ⊢ (∅ ∈ (ω ∖ {∅}) ↔ (∅ ∈ ω ∧ ¬ ∅ ∈ {∅}))
28	26, 27	sylib 173	. . . . . . . . 9 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ∧ ¬ ∅ ∈ {∅}))
29	28	pm3.27d 262	. . . . . . . 8 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅})
30	23, 29	sylbi 174	. . . . . . 7 ⊢ (¬ (ω ∖ {∅}) ⊂ ω → ¬ ∅ ∈ {∅})
31	30	a3i 69	. . . . . 6 ⊢ (∅ ∈ {∅} → (ω ∖ {∅}) ⊂ ω)
32	20, 31	ax-mp 6	. . . . 5 ⊢ (ω ∖ {∅}) ⊂ ω
33		unidif0 1944	. . . . . . 7 ⊢ ∪(ω ∖ {∅}) = ∪ω
34		limom 2387	. . . . . . . 8 ⊢ Lim ω
35		limuni 2284	. . . . . . . 8 ⊢ (Lim ω → ω = ∪ω)
36	34, 35	ax-mp 6	. . . . . . 7 ⊢ ω = ∪ω
37	33, 36	eqtr4 1122	. . . . . 6 ⊢ ∪(ω ∖ {∅}) = ω
38	37	psseq2i 1562	. . . . 5 ⊢ ((ω ∖ {∅}) ⊂ ∪(ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊂ ω)
39	32, 38	mpbir 165	. . . 4 ⊢ (ω ∖ {∅}) ⊂ ∪(ω ∖ {∅})
40		psseq1 1559	. . . . . 6 ⊢ (x = (ω ∖ {∅}) → (x ⊂ ∪x ↔ (ω ∖ {∅}) ⊂ ∪x))
41		unieq 1927	. . . . . . 7 ⊢ (x = (ω ∖ {∅}) → ∪x = ∪(ω ∖ {∅}))
42	41	psseq2d 1565	. . . . . 6 ⊢ (x = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊂ ∪x ↔ (ω ∖ {∅}) ⊂ ∪(ω ∖ {∅})))
43	40, 42	bitrd 406	. . . . 5 ⊢ (x = (ω ∖ {∅}) → (x ⊂ ∪x ↔ (ω ∖ {∅}) ⊂ ∪(ω ∖ {∅})))
44	43	cla4egv 1397	. . . 4 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊂ ∪(ω ∖ {∅}) → ∃x x ⊂ ∪x))
45	39, 44	mpi 44	. . 3 ⊢ ((ω ∖ {∅}) ∈ V → ∃x x ⊂ ∪x)
46	18, 45	syl 12	. 2 ⊢ (ω ∈ V → ∃x x ⊂ ∪x)
47	17, 46	impbi 139	1 ⊢ (∃x x ⊂ ∪x ↔ ω ∈ V)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487   ⊂ wpss 1488  ∅c0 1707  {csn 1808  ∪cuni 1919  {copab 2055  Lim wlim 2200  ωcom 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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