Theorem aceq5lem2 3559

Description: Lemma for aceq5 3563.

Hypothesis

Ref	Expression
aceq5lem.1	⊢ A = {u∣(¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))}

Assertion

Ref	Expression
aceq5lem2	⊢ (⟨w, g⟩ ∈ ∪A ↔ (w ∈ h ∧ g ∈ w))

Distinct variable group(s):   w,u,t,h,g   w,A,g

Proof of Theorem aceq5lem2

Step	Hyp	Ref	Expression
1		aceq5lem.1	. . . 4 ⊢ A = {u∣(¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))}
2	1	unieqi 1928	. . 3 ⊢ ∪A = ∪{u∣(¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))}
3	2	eleq2i 1153	. 2 ⊢ (⟨w, g⟩ ∈ ∪A ↔ ⟨w, g⟩ ∈ ∪{u∣(¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))})
4		eluniab 1926	. . 3 ⊢ (⟨w, g⟩ ∈ ∪{u∣(¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))} ↔ ∃u(⟨w, g⟩ ∈ u ∧ (¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))))
5		r19.42v 1303	. . . . 5 ⊢ (∃t ∈ h ((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ ((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ ∃t ∈ h u = ({t} × t)))
6		anass 336	. . . . 5 ⊢ (((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ ∃t ∈ h u = ({t} × t)) ↔ (⟨w, g⟩ ∈ u ∧ (¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))))
7	5, 6	bitr2 152	. . . 4 ⊢ ((⟨w, g⟩ ∈ u ∧ (¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))) ↔ ∃t ∈ h ((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)))
8	7	biex 733	. . 3 ⊢ (∃u(⟨w, g⟩ ∈ u ∧ (¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))) ↔ ∃u∃t ∈ h ((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)))
9		rexcom4 1361	. . . 4 ⊢ (∃t ∈ h ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ ∃u∃t ∈ h ((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)))
10		df-rex 1206	. . . 4 ⊢ (∃t ∈ h ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ ∃t(t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))))
11	9, 10	bitr3 153	. . 3 ⊢ (∃u∃t ∈ h ((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ ∃t(t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))))
12	4, 8, 11	3bitr 155	. 2 ⊢ (⟨w, g⟩ ∈ ∪{u∣(¬ u = ∅ ∧ ∃t ∈ h u = ({t} × t))} ↔ ∃t(t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))))
13		ancom 333	. . . . . . . . 9 ⊢ (((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ (u = ({t} × t) ∧ (⟨w, g⟩ ∈ u ∧ ¬ u = ∅)))
14		n0i 1712	. . . . . . . . . . 11 ⊢ (⟨w, g⟩ ∈ u → ¬ u = ∅)
15	14	pm4.71i 483	. . . . . . . . . 10 ⊢ (⟨w, g⟩ ∈ u ↔ (⟨w, g⟩ ∈ u ∧ ¬ u = ∅))
16	15	anbi2i 367	. . . . . . . . 9 ⊢ ((u = ({t} × t) ∧ ⟨w, g⟩ ∈ u) ↔ (u = ({t} × t) ∧ (⟨w, g⟩ ∈ u ∧ ¬ u = ∅)))
17	13, 16	bitr4 154	. . . . . . . 8 ⊢ (((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ (u = ({t} × t) ∧ ⟨w, g⟩ ∈ u))
18	17	biex 733	. . . . . . 7 ⊢ (∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ ∃u(u = ({t} × t) ∧ ⟨w, g⟩ ∈ u))
19		snex 1859	. . . . . . . . 9 ⊢ {t} ∈ V
20		visset 1350	. . . . . . . . 9 ⊢ t ∈ V
21	19, 20	xpex 2488	. . . . . . . 8 ⊢ ({t} × t) ∈ V
22		eleq2 1150	. . . . . . . 8 ⊢ (u = ({t} × t) → (⟨w, g⟩ ∈ u ↔ ⟨w, g⟩ ∈ ({t} × t)))
23	21, 22	ceqsexv 1371	. . . . . . 7 ⊢ (∃u(u = ({t} × t) ∧ ⟨w, g⟩ ∈ u) ↔ ⟨w, g⟩ ∈ ({t} × t))
24	18, 23	bitr 151	. . . . . 6 ⊢ (∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t)) ↔ ⟨w, g⟩ ∈ ({t} × t))
25	24	anbi2i 367	. . . . 5 ⊢ ((t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))) ↔ (t ∈ h ∧ ⟨w, g⟩ ∈ ({t} × t)))
26		visset 1350	. . . . . . . 8 ⊢ g ∈ V
27	26	opelxp 2452	. . . . . . 7 ⊢ (⟨w, g⟩ ∈ ({t} × t) ↔ (w ∈ {t} ∧ g ∈ t))
28		elsn 1820	. . . . . . . . 9 ⊢ (w ∈ {t} ↔ w = t)
29		cleqcom 1103	. . . . . . . . 9 ⊢ (w = t ↔ t = w)
30	28, 29	bitr 151	. . . . . . . 8 ⊢ (w ∈ {t} ↔ t = w)
31	30	anbi1i 368	. . . . . . 7 ⊢ ((w ∈ {t} ∧ g ∈ t) ↔ (t = w ∧ g ∈ t))
32	27, 31	bitr 151	. . . . . 6 ⊢ (⟨w, g⟩ ∈ ({t} × t) ↔ (t = w ∧ g ∈ t))
33	32	anbi2i 367	. . . . 5 ⊢ ((t ∈ h ∧ ⟨w, g⟩ ∈ ({t} × t)) ↔ (t ∈ h ∧ (t = w ∧ g ∈ t)))
34		an12 370	. . . . 5 ⊢ ((t ∈ h ∧ (t = w ∧ g ∈ t)) ↔ (t = w ∧ (t ∈ h ∧ g ∈ t)))
35	25, 33, 34	3bitr 155	. . . 4 ⊢ ((t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))) ↔ (t = w ∧ (t ∈ h ∧ g ∈ t)))
36	35	biex 733	. . 3 ⊢ (∃t(t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))) ↔ ∃t(t = w ∧ (t ∈ h ∧ g ∈ t)))
37		visset 1350	. . . 4 ⊢ w ∈ V
38		eleq1 1149	. . . . 5 ⊢ (t = w → (t ∈ h ↔ w ∈ h))
39		eleq2 1150	. . . . 5 ⊢ (t = w → (g ∈ t ↔ g ∈ w))
40	38, 39	anbi12d 476	. . . 4 ⊢ (t = w → ((t ∈ h ∧ g ∈ t) ↔ (w ∈ h ∧ g ∈ w)))
41	37, 40	ceqsexv 1371	. . 3 ⊢ (∃t(t = w ∧ (t ∈ h ∧ g ∈ t)) ↔ (w ∈ h ∧ g ∈ w))
42	36, 41	bitr 151	. 2 ⊢ (∃t(t ∈ h ∧ ∃u((⟨w, g⟩ ∈ u ∧ ¬ u = ∅) ∧ u = ({t} × t))) ↔ (w ∈ h ∧ g ∈ w))
43	3, 12, 42	3bitr 155	1 ⊢ (⟨w, g⟩ ∈ ∪A ↔ (w ∈ h ∧ g ∈ w))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∅c0 1707  {csn 1808  ⟨cop 1810  ∪cuni 1919   × cxp 2408

This theorem is referenced by:  aceq5lem5 3562

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-opab 2098  df-xp 2424  df-rel 2425

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