Theorem ord3ex 5376

Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7719. (Contributed by NM, 2-May-2009.)

Assertion

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

Proof of Theorem ord3ex

Step	Hyp	Ref	Expression
1		df-tp 4626	. 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2		pwpr 4894	. . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3		pp0ex 5375	. . . . 5 ⊢ {∅, {∅}} ∈ V
4	3	pwex 5369	. . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V
5	2, 4	eqeltrri 2822	. . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6		snsspr2 4811	. . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7		unss2 4174	. . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
8	6, 7	ax-mp 5	. . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
9	5, 8	ssexi 5313	. 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
10	1, 9	eqeltri 2821	1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Syntax hints:   ∈ wcel 2098  Vcvv 3466   ∪ cun 3939   ⊆ wss 3941  ∅c0 4315  𝒫 cpw 4595  {csn 4621  {cpr 4623  {ctp 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626

This theorem is referenced by: (None)