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Mirrors > Home > MPE Home > Th. List > ord3ex | Structured version Visualization version GIF version |
Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7753. (Contributed by NM, 2-May-2009.) |
Ref | Expression |
---|---|
ord3ex | ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4635 | . 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
2 | pwpr 4905 | . . . 4 ⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) | |
3 | pp0ex 5391 | . . . . 5 ⊢ {∅, {∅}} ∈ V | |
4 | 3 | pwex 5385 | . . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V |
5 | 2, 4 | eqeltrri 2835 | . . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V |
6 | snsspr2 4819 | . . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} | |
7 | unss2 4196 | . . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) |
9 | 5, 8 | ssexi 5327 | . 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V |
10 | 1, 9 | eqeltri 2834 | 1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3477 ∪ cun 3960 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 {csn 4630 {cpr 4632 {ctp 4634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 |
This theorem is referenced by: (None) |
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