Theorem inf3lem3 3466

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 3471 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 3447.

Hypotheses

Ref	Expression
inf3lem.1	⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
inf3lem.2	⊢ F = (rec(G, ∅) ↾ ω)
inf3lem.3	⊢ A ∈ V
inf3lem.4	⊢ B ∈ V

Assertion

Ref	Expression
inf3lem3	⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → ¬ (F ‘A) = (F ‘suc A)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lem3

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . . 7 ⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
2		inf3lem.2	. . . . . . 7 ⊢ F = (rec(G, ∅) ↾ ω)
3		inf3lem.3	. . . . . . 7 ⊢ A ∈ V
4		inf3lem.4	. . . . . . 7 ⊢ B ∈ V
5	1, 2, 3, 4	inf3lemd 3463	. . . . . 6 ⊢ (A ∈ ω → (F ‘A) ⊆ x)
6	5	a1d 14	. . . . 5 ⊢ (A ∈ ω → ((¬ x = ∅ ∧ x ⊆ ∪x) → (F ‘A) ⊆ x))
7	1, 2, 3, 4	inf3lem2 3465	. . . . . 6 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → ¬ (F ‘A) = x))
8	7	com12 13	. . . . 5 ⊢ (A ∈ ω → ((¬ x = ∅ ∧ x ⊆ ∪x) → ¬ (F ‘A) = x))
9	6, 8	jcad 455	. . . 4 ⊢ (A ∈ ω → ((¬ x = ∅ ∧ x ⊆ ∪x) → ((F ‘A) ⊆ x ∧ ¬ (F ‘A) = x)))
10		pssdifn0 1750	. . . 4 ⊢ (((F ‘A) ⊆ x ∧ ¬ (F ‘A) = x) → ¬ (x ∖ (F ‘A)) = ∅)
11	9, 10	syl6 23	. . 3 ⊢ (A ∈ ω → ((¬ x = ∅ ∧ x ⊆ ∪x) → ¬ (x ∖ (F ‘A)) = ∅))
12	1, 2, 3, 4	inf3lemc 3462	. . . . . . . . . 10 ⊢ (A ∈ ω → (F ‘suc A) = (G ‘(F ‘A)))
13	12	eleq2d 1156	. . . . . . . . 9 ⊢ (A ∈ ω → (v ∈ (F ‘suc A) ↔ v ∈ (G ‘(F ‘A))))
14		eldifi 1591	. . . . . . . . . . 11 ⊢ (v ∈ (x ∖ (F ‘A)) → v ∈ x)
15		inssdif0 1754	. . . . . . . . . . . 12 ⊢ ((v ∩ x) ⊆ (F ‘A) ↔ (v ∩ (x ∖ (F ‘A))) = ∅)
16	15	biimpr 134	. . . . . . . . . . 11 ⊢ ((v ∩ (x ∖ (F ‘A))) = ∅ → (v ∩ x) ⊆ (F ‘A))
17	14, 16	anim12i 268	. . . . . . . . . 10 ⊢ ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → (v ∈ x ∧ (v ∩ x) ⊆ (F ‘A)))
18		visset 1350	. . . . . . . . . . 11 ⊢ v ∈ V
19		fvex 2838	. . . . . . . . . . 11 ⊢ (F ‘A) ∈ V
20	1, 2, 18, 19	inf3lema 3460	. . . . . . . . . 10 ⊢ (v ∈ (G ‘(F ‘A)) ↔ (v ∈ x ∧ (v ∩ x) ⊆ (F ‘A)))
21	17, 20	sylibr 175	. . . . . . . . 9 ⊢ ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → v ∈ (G ‘(F ‘A)))
22	13, 21	syl5bir 184	. . . . . . . 8 ⊢ (A ∈ ω → ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → v ∈ (F ‘suc A)))
23		eldifn 1592	. . . . . . . . . 10 ⊢ (v ∈ (x ∖ (F ‘A)) → ¬ v ∈ (F ‘A))
24	23	adantr 306	. . . . . . . . 9 ⊢ ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → ¬ v ∈ (F ‘A))
25	24	a1i 7	. . . . . . . 8 ⊢ (A ∈ ω → ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → ¬ v ∈ (F ‘A)))
26	22, 25	jcad 455	. . . . . . 7 ⊢ (A ∈ ω → ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → (v ∈ (F ‘suc A) ∧ ¬ v ∈ (F ‘A))))
27		eleq2 1150	. . . . . . . . . 10 ⊢ ((F ‘A) = (F ‘suc A) → (v ∈ (F ‘A) ↔ v ∈ (F ‘suc A)))
28	27	biimprd 136	. . . . . . . . 9 ⊢ ((F ‘A) = (F ‘suc A) → (v ∈ (F ‘suc A) → v ∈ (F ‘A)))
29		iman 205	. . . . . . . . 9 ⊢ ((v ∈ (F ‘suc A) → v ∈ (F ‘A)) ↔ ¬ (v ∈ (F ‘suc A) ∧ ¬ v ∈ (F ‘A)))
30	28, 29	sylib 173	. . . . . . . 8 ⊢ ((F ‘A) = (F ‘suc A) → ¬ (v ∈ (F ‘suc A) ∧ ¬ v ∈ (F ‘A)))
31	30	con2i 89	. . . . . . 7 ⊢ ((v ∈ (F ‘suc A) ∧ ¬ v ∈ (F ‘A)) → ¬ (F ‘A) = (F ‘suc A))
32	26, 31	syl6 23	. . . . . 6 ⊢ (A ∈ ω → ((v ∈ (x ∖ (F ‘A)) ∧ (v ∩ (x ∖ (F ‘A))) = ∅) → ¬ (F ‘A) = (F ‘suc A)))
33	32	exp3a 292	. . . . 5 ⊢ (A ∈ ω → (v ∈ (x ∖ (F ‘A)) → ((v ∩ (x ∖ (F ‘A))) = ∅ → ¬ (F ‘A) = (F ‘suc A))))
34	33	r19.23adv 1286	. . . 4 ⊢ (A ∈ ω → (∃v ∈ (x ∖ (F ‘A))(v ∩ (x ∖ (F ‘A))) = ∅ → ¬ (F ‘A) = (F ‘suc A)))
35		visset 1350	. . . . . 6 ⊢ x ∈ V
36		difss 1596	. . . . . 6 ⊢ (x ∖ (F ‘A)) ⊆ x
37	35, 36	ssexi 1701	. . . . 5 ⊢ (x ∖ (F ‘A)) ∈ V
38	37	zfreg 3447	. . . 4 ⊢ (¬ (x ∖ (F ‘A)) = ∅ → ∃v ∈ (x ∖ (F ‘A))(v ∩ (x ∖ (F ‘A))) = ∅)
39	34, 38	syl5 22	. . 3 ⊢ (A ∈ ω → (¬ (x ∖ (F ‘A)) = ∅ → ¬ (F ‘A) = (F ‘suc A)))
40	11, 39	syld 27	. 2 ⊢ (A ∈ ω → ((¬ x = ∅ ∧ x ⊆ ∪x) → ¬ (F ‘A) = (F ‘suc A)))
41	40	com12 13	1 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → ¬ (F ‘A) = (F ‘suc A)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  Vcvv 1348   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919  {copab 2055  suc csuc 2201  ωcom 2372   ↾ cres 2412   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  inf3lem4 3467

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-fv 2438  df-rdg 2970

metamath.org