Theorem unon 2338

Description: The class of all ordinals is its own union. Exercise 11 of [TakeutiZaring] p. 40.

Assertion

Ref	Expression
unon	⊢ ∪On = On

Proof of Theorem unon

Step	Hyp	Ref	Expression
1		eluni2 1923	. . . 4 ⊢ (x ∈ ∪On ↔ ∃y ∈ On x ∈ y)
2		onelon 2223	. . . . . 6 ⊢ ((y ∈ On ∧ x ∈ y) → x ∈ On)
3	2	exp 291	. . . . 5 ⊢ (y ∈ On → (x ∈ y → x ∈ On))
4	3	r19.23aiv 1284	. . . 4 ⊢ (∃y ∈ On x ∈ y → x ∈ On)
5	1, 4	sylbi 174	. . 3 ⊢ (x ∈ ∪On → x ∈ On)
6		suceloni 2314	. . . . 5 ⊢ (x ∈ On → suc x ∈ On)
7		visset 1350	. . . . . 6 ⊢ x ∈ V
8	7	sucid 2304	. . . . 5 ⊢ x ∈ suc x
9	6, 8	jctil 240	. . . 4 ⊢ (x ∈ On → (x ∈ suc x ∧ suc x ∈ On))
10		elunii 1924	. . . 4 ⊢ ((x ∈ suc x ∧ suc x ∈ On) → x ∈ ∪On)
11	9, 10	syl 12	. . 3 ⊢ (x ∈ On → x ∈ ∪On)
12	5, 11	impbi 139	. 2 ⊢ (x ∈ ∪On ↔ x ∈ On)
13	12	cleqri 1101	1 ⊢ ∪On = On

Syntax hints:   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∪cuni 1919  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  ordunisuc 2339  limon 2342  orduninsuc 2365

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-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  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-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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