Theorem ord3ex 5385

Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7729. (Contributed by NM, 2-May-2009.)

Assertion

Ref	Expression
ord3ex	⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex

Step	Hyp	Ref	Expression
1		df-tp 4633	. 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2		pwpr 4902	. . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3		pp0ex 5384	. . . . 5 ⊢ {∅, {∅}} ∈ V
4	3	pwex 5378	. . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V
5	2, 4	eqeltrri 2829	. . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6		snsspr2 4818	. . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7		unss2 4181	. . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
8	6, 7	ax-mp 5	. . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
9	5, 8	ssexi 5322	. 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
10	1, 9	eqeltri 2828	1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3473   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628  {cpr 4630  {ctp 4632

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633

This theorem is referenced by: (None)