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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 4091 which more directly illustrates the contrast with el2oss1o 13115. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
ss1oel2o | ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4091 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | |
2 | df1o2 6294 | . . . . 5 ⊢ 1o = {∅} | |
3 | 2 | sseq2i 3094 | . . . 4 ⊢ (𝑥 ⊆ 1o ↔ 𝑥 ⊆ {∅}) |
4 | df2o2 6296 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
5 | 4 | eleq2i 2184 | . . . . 5 ⊢ (𝑥 ∈ 2o ↔ 𝑥 ∈ {∅, {∅}}) |
6 | vex 2663 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | elpr 3518 | . . . . 5 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})) |
8 | 5, 7 | bitri 183 | . . . 4 ⊢ (𝑥 ∈ 2o ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})) |
9 | 3, 8 | imbi12i 238 | . . 3 ⊢ ((𝑥 ⊆ 1o → 𝑥 ∈ 2o) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) |
10 | 9 | albii 1431 | . 2 ⊢ (∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) |
11 | 1, 10 | bitr4i 186 | 1 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 682 ∀wal 1314 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 ∅c0 3333 {csn 3497 {cpr 3498 EXMIDwem 4088 1oc1o 6274 2oc2o 6275 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 df-exmid 4089 df-suc 4263 df-1o 6281 df-2o 6282 |
This theorem is referenced by: (None) |
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