![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > exmid1dc | GIF version |
Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4192 or ordtriexmid 4519. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.) |
Ref | Expression |
---|---|
exmid1dc.x | ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅}) |
Ref | Expression |
---|---|
exmid1dc | ⊢ (𝜑 → EXMID) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid1dc.x | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅}) | |
2 | exmiddc 836 | . . . . . . 7 ⊢ (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅})) | |
3 | 1, 2 | syl 14 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅})) |
4 | df-ne 2348 | . . . . . . . . 9 ⊢ (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅}) | |
5 | pwntru 4198 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅) | |
6 | 5 | ex 115 | . . . . . . . . 9 ⊢ (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅)) |
7 | 4, 6 | biimtrrid 153 | . . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅)) |
8 | 7 | orim2d 788 | . . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))) |
9 | 8 | adantl 277 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))) |
10 | 3, 9 | mpd 13 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)) |
11 | 10 | orcomd 729 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅})) |
12 | 11 | ex 115 | . . 3 ⊢ (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) |
13 | 12 | alrimiv 1874 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) |
14 | exmid01 4197 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | |
15 | 13, 14 | sylibr 134 | 1 ⊢ (𝜑 → EXMID) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 ∀wal 1351 = wceq 1353 ≠ wne 2347 ⊆ wss 3129 ∅c0 3422 {csn 3592 EXMIDwem 4193 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-nul 4128 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3598 df-exmid 4194 |
This theorem is referenced by: pw1fin 6907 exmidonfin 7190 exmidaclem 7204 exmidontri 7235 exmidontri2or 7239 |
Copyright terms: Public domain | W3C validator |