Theorem ord3ex 5337

Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7691. (Contributed by NM, 2-May-2009.)

Assertion

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

Proof of Theorem ord3ex

Step	Hyp	Ref	Expression
1		df-tp 4590	. 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2		pwpr 4861	. . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3		pp0ex 5336	. . . . 5 ⊢ {∅, {∅}} ∈ V
4	3	pwex 5330	. . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V
5	2, 4	eqeltrri 2825	. . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6		snsspr2 4775	. . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7		unss2 4146	. . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
8	6, 7	ax-mp 5	. . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
9	5, 8	ssexi 5272	. 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
10	1, 9	eqeltri 2824	1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  {cpr 4587  {ctp 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590

This theorem is referenced by: (None)