Theorem unon 7830

Description: The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)

Assertion

Ref	Expression
unon	⊢ ∪ On = On

Proof of Theorem unon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4892	. . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦)
2		onelon 6382	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
3	2	rexlimiva 3134	. . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On)
4	1, 3	sylbi 217	. . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On)
5		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
6	5	sucid 6441	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
7		onsuc 7810	. . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		elunii 4893	. . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On)
9	6, 7, 8	sylancr 587	. . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On)
10	4, 9	impbii 209	. 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On)
11	10	eqriv 2733	1 ⊢ ∪ On = On

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  ∪ cuni 4888  Oncon0 6357  suc csuc 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361  df-suc 6363

This theorem is referenced by:  ordunisuc  7831  limon  7835  orduninsuc  7843  ordtoplem  36458  ordcmp  36470  onsupnmax  43219