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Mirrors > Home > ILE Home > Th. List > 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 14026. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
el2oss1o | ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 3606 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | df2o3 6409 | . . 3 ⊢ 2o = {∅, 1o} | |
3 | 1, 2 | eleq2s 2265 | . 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
4 | 0ss 3453 | . . . 4 ⊢ ∅ ⊆ 1o | |
5 | sseq1 3170 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o)) | |
6 | 4, 5 | mpbiri 167 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o) |
7 | eqimss 3201 | . . 3 ⊢ (𝐴 = 1o → 𝐴 ⊆ 1o) | |
8 | 6, 7 | jaoi 711 | . 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 703 = wceq 1348 ∈ wcel 2141 ⊆ wss 3121 ∅c0 3414 {cpr 3584 1oc1o 6388 2oc2o 6389 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3589 df-pr 3590 df-suc 4356 df-1o 6395 df-2o 6396 |
This theorem is referenced by: nnnninfeq2 7105 nninfwlpoimlemg 7151 nninfsellemsuc 14045 |
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