Theorem unon 7194

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 4590	. . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦)
2		onelon 5907	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
3	2	rexlimiva 3164	. . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On)
4	1, 3	sylbi 207	. . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On)
5		vex 3341	. . . . 5 ⊢ 𝑥 ∈ V
6	5	sucid 5963	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
7		suceloni 7176	. . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		elunii 4591	. . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On)
9	6, 7, 8	sylancr 698	. . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On)
10	4, 9	impbii 199	. 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On)
11	10	eqriv 2755	1 ⊢ ∪ On = On

Syntax hints:   = wceq 1630   ∈ wcel 2137  ∃wrex 3049  ∪ cuni 4586  Oncon0 5882  suc csuc 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-tr 4903  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-ord 5885  df-on 5886  df-suc 5888

This theorem is referenced by:  ordunisuc  7195  limon  7199  orduninsuc  7206  ordtoplem  32738  ordcmp  32750