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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 4129 which more directly illustrates the contrast with el2oss1o 13359. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
ss1oel2o | ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4129 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | |
2 | df1o2 6334 | . . . . 5 ⊢ 1o = {∅} | |
3 | 2 | sseq2i 3129 | . . . 4 ⊢ (𝑥 ⊆ 1o ↔ 𝑥 ⊆ {∅}) |
4 | df2o2 6336 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
5 | 4 | eleq2i 2207 | . . . . 5 ⊢ (𝑥 ∈ 2o ↔ 𝑥 ∈ {∅, {∅}}) |
6 | vex 2692 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | elpr 3553 | . . . . 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 1447 | . 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 698 ∀wal 1330 = wceq 1332 ∈ wcel 1481 ⊆ wss 3076 ∅c0 3368 {csn 3532 {cpr 3533 EXMIDwem 4126 1oc1o 6314 2oc2o 6315 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-exmid 4127 df-suc 4301 df-1o 6321 df-2o 6322 |
This theorem is referenced by: (None) |
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