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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 3540 | . 2 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}}) | |
2 | pp0ex 4121 | . . . . 5 ⊢ {∅, {∅}} ∈ V | |
3 | 2 | pwex 4115 | . . . 4 ⊢ 𝒫 {∅, {∅}} ∈ V |
4 | pwprss 3740 | . . . 4 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}} | |
5 | 3, 4 | ssexi 4074 | . . 3 ⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V |
6 | snsspr2 3677 | . . . 4 ⊢ {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} | |
7 | unss2 3252 | . . . 4 ⊢ ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) |
9 | 5, 8 | ssexi 4074 | . 2 ⊢ ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V |
10 | 1, 9 | eqeltri 2213 | 1 ⊢ {∅, {∅}, {∅, {∅}}} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 ⊆ wss 3076 ∅c0 3368 𝒫 cpw 3515 {csn 3532 {cpr 3533 {ctp 3534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 |
This theorem is referenced by: (None) |
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