Theorem unon 7836

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 4881	. . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦)
2		onelon 6392	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
3	2	rexlimiva 3161	. . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On)
4	1, 3	sylbi 220	. . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On)
5		vex 3462	. . . . 5 ⊢ 𝑥 ∈ V
6	5	sucid 6452	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
7		onsuc 7818	. . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		elunii 4882	. . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On)
9	6, 7, 8	sylancr 599	. . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On)
10	4, 9	impbii 212	. 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On)
11	10	eqriv 2763	1 ⊢ ∪ On = On

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  ∪ cuni 4877  Oncon0 6367  suc csuc 6369

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370  df-on 6371  df-suc 6373

This theorem is used by:  ordunisuc  7837  limon  7841  orduninsuc  7848  ordtoplem  36987  ordcmp  36999  onsupnmax  43996