Theorem unon 4632

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 3917	. . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦)
2		onelon 4504	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
3	2	rexlimiva 2655	. . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On)
4	1, 3	sylbi 121	. . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On)
5		vex 2815	. . . . 5 ⊢ 𝑥 ∈ V
6	5	sucid 4537	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
7		onsuc 4622	. . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		elunii 3918	. . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On)
9	6, 7, 8	sylancr 414	. . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On)
10	4, 9	impbii 126	. 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On)
11	10	eqriv 2229	1 ⊢ ∪ On = On

Syntax hints:   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  ∪ cuni 3913  Oncon0 4483  suc csuc 4485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488  df-suc 4491

This theorem is referenced by:  limon  4634  onintonm  4638  tfri1dALT  6581  rdgon  6616