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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > ss1oel2o | GIF version |
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4210 which more directly illustrates the contrast with el2oss1o 6458. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
ss1oel2o | ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4210 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | |
2 | df1o2 6444 | . . . . 5 ⊢ 1o = {∅} | |
3 | 2 | sseq2i 3194 | . . . 4 ⊢ (𝑥 ⊆ 1o ↔ 𝑥 ⊆ {∅}) |
4 | df2o2 6446 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
5 | 4 | eleq2i 2254 | . . . . 5 ⊢ (𝑥 ∈ 2o ↔ 𝑥 ∈ {∅, {∅}}) |
6 | vex 2752 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | elpr 3625 | . . . . 5 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})) |
8 | 5, 7 | bitri 184 | . . . 4 ⊢ (𝑥 ∈ 2o ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})) |
9 | 3, 8 | imbi12i 239 | . . 3 ⊢ ((𝑥 ⊆ 1o → 𝑥 ∈ 2o) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) |
10 | 9 | albii 1480 | . 2 ⊢ (∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) |
11 | 1, 10 | bitr4i 187 | 1 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 709 ∀wal 1361 = wceq 1363 ∈ wcel 2158 ⊆ wss 3141 ∅c0 3434 {csn 3604 {cpr 3605 EXMIDwem 4206 1oc1o 6424 2oc2o 6425 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-nul 4141 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-sn 3610 df-pr 3611 df-exmid 4207 df-suc 4383 df-1o 6431 df-2o 6432 |
This theorem is referenced by: (None) |
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