Theorem unon 7548

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 4844	. . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦)
2		onelon 6218	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
3	2	rexlimiva 3283	. . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On)
4	1, 3	sylbi 219	. . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On)
5		vex 3499	. . . . 5 ⊢ 𝑥 ∈ V
6	5	sucid 6272	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
7		suceloni 7530	. . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		elunii 4845	. . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On)
9	6, 7, 8	sylancr 589	. . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On)
10	4, 9	impbii 211	. 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On)
11	10	eqriv 2820	1 ⊢ ∪ On = On

Syntax hints:   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  ∪ cuni 4840  Oncon0 6193  suc csuc 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-suc 6199

This theorem is referenced by:  ordunisuc  7549  limon  7553  orduninsuc  7560  ordtoplem  33785  ordcmp  33797