Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > el2oss1o | GIF version |
Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13178. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
el2oss1o | ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 3545 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | df2o3 6320 | . . 3 ⊢ 2o = {∅, 1o} | |
3 | 1, 2 | eleq2s 2232 | . 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
4 | 0ss 3396 | . . . 4 ⊢ ∅ ⊆ 1o | |
5 | sseq1 3115 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o)) | |
6 | 4, 5 | mpbiri 167 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o) |
7 | eqimss 3146 | . . 3 ⊢ (𝐴 = 1o → 𝐴 ⊆ 1o) | |
8 | 6, 7 | jaoi 705 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → 𝐴 ⊆ 1o) |
9 | 3, 8 | syl 14 | 1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ⊆ wss 3066 ∅c0 3358 {cpr 3523 1oc1o 6299 2oc2o 6300 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-pr 3529 df-suc 4288 df-1o 6306 df-2o 6307 |
This theorem is referenced by: nninfsellemsuc 13197 |
Copyright terms: Public domain | W3C validator |