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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 14746. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
el2oss1o | ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 3616 | . . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
2 | df2o3 6431 | . . 3 ⊢ 2o = {∅, 1o} | |
3 | 1, 2 | eleq2s 2272 | . 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
4 | 0ss 3462 | . . . 4 ⊢ ∅ ⊆ 1o | |
5 | sseq1 3179 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o)) | |
6 | 4, 5 | mpbiri 168 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o) |
7 | eqimss 3210 | . . 3 ⊢ (𝐴 = 1o → 𝐴 ⊆ 1o) | |
8 | 6, 7 | jaoi 716 | . 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 708 = wceq 1353 ∈ wcel 2148 ⊆ wss 3130 ∅c0 3423 {cpr 3594 1oc1o 6410 2oc2o 6411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2740 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-sn 3599 df-pr 3600 df-suc 4372 df-1o 6417 df-2o 6418 |
This theorem is referenced by: nnnninfeq2 7127 nninfwlpoimlemg 7173 nninfsellemsuc 14764 |
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