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Mirrors > Home > ILE Home > Th. List > ord3ex | GIF version |
Description: The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.) |
Ref | Expression |
---|---|
ord3ex | ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3406 | . 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
2 | pp0ex 3960 | . . . . 5 ⊢ {∅, {∅}} ∈ V | |
3 | 2 | pwex 3953 | . . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V |
4 | pwprss 3597 | . . . 4 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}} | |
5 | 3, 4 | ssexi 3916 | . . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V |
6 | snsspr2 3534 | . . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} | |
7 | unss2 3143 | . . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})) | |
8 | 6, 7 | ax-mp 7 | . . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) |
9 | 5, 8 | ssexi 3916 | . 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V |
10 | 1, 9 | eqeltri 2151 | 1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 Vcvv 2601 ∪ cun 2971 ⊆ wss 2973 ∅c0 3251 𝒫 cpw 3382 {csn 3398 {cpr 3399 {ctp 3400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-tp 3406 |
This theorem is referenced by: (None) |
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