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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 4038 which more directly illustrates the contrast with el2oss1o 12160. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
ss1oel2o | ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4038 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | |
2 | df1o2 6208 | . . . . 5 ⊢ 1o = {∅} | |
3 | 2 | sseq2i 3052 | . . . 4 ⊢ (𝑥 ⊆ 1o ↔ 𝑥 ⊆ {∅}) |
4 | df2o2 6210 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
5 | 4 | eleq2i 2155 | . . . . 5 ⊢ (𝑥 ∈ 2o ↔ 𝑥 ∈ {∅, {∅}}) |
6 | vex 2623 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | elpr 3471 | . . . . 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 1405 | . 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 665 ∀wal 1288 = wceq 1290 ∈ wcel 1439 ⊆ wss 3000 ∅c0 3287 {csn 3450 {cpr 3451 EXMIDwem 4035 1oc1o 6188 2oc2o 6189 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-nul 3971 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-sn 3456 df-pr 3457 df-exmid 4036 df-suc 4207 df-1o 6195 df-2o 6196 |
This theorem is referenced by: (None) |
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