Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
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 4182 which more directly illustrates the contrast with el2oss1o 6420. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
ss1oel2o | ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4182 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | |
2 | df1o2 6406 | . . . . 5 ⊢ 1o = {∅} | |
3 | 2 | sseq2i 3174 | . . . 4 ⊢ (𝑥 ⊆ 1o ↔ 𝑥 ⊆ {∅}) |
4 | df2o2 6408 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
5 | 4 | eleq2i 2237 | . . . . 5 ⊢ (𝑥 ∈ 2o ↔ 𝑥 ∈ {∅, {∅}}) |
6 | vex 2733 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | 6 | elpr 3602 | . . . . 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 1463 | . 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 703 ∀wal 1346 = wceq 1348 ∈ wcel 2141 ⊆ wss 3121 ∅c0 3414 {csn 3581 {cpr 3582 EXMIDwem 4178 1oc1o 6386 2oc2o 6387 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4113 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3587 df-pr 3588 df-exmid 4179 df-suc 4354 df-1o 6393 df-2o 6394 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |