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Mirrors > Home > ILE Home > Th. List > ordtriexmidlem2 | GIF version |
Description: Lemma for decidability and ordinals. The set {x ∈ {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4210 or weak linearity in ordsoexmid 4240) with a proposition φ. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Ref | Expression |
---|---|
ordtriexmidlem2 | ⊢ ({x ∈ {∅} ∣ φ} = ∅ → ¬ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3222 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | eleq2 2098 | . . 3 ⊢ ({x ∈ {∅} ∣ φ} = ∅ → (∅ ∈ {x ∈ {∅} ∣ φ} ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | mtbiri 599 | . 2 ⊢ ({x ∈ {∅} ∣ φ} = ∅ → ¬ ∅ ∈ {x ∈ {∅} ∣ φ}) |
4 | 0ex 3875 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snid 3394 | . . 3 ⊢ ∅ ∈ {∅} |
6 | biidd 161 | . . . 4 ⊢ (x = ∅ → (φ ↔ φ)) | |
7 | 6 | elrab3 2693 | . . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {x ∈ {∅} ∣ φ} ↔ φ)) |
8 | 5, 7 | ax-mp 7 | . 2 ⊢ (∅ ∈ {x ∈ {∅} ∣ φ} ↔ φ) |
9 | 3, 8 | sylnib 600 | 1 ⊢ ({x ∈ {∅} ∣ φ} = ∅ → ¬ φ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {crab 2304 ∅c0 3218 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-v 2553 df-dif 2914 df-nul 3219 df-sn 3373 |
This theorem is referenced by: ordtriexmid 4210 ordtri2orexmid 4211 onsucsssucexmid 4212 ordsoexmid 4240 ordpwsucexmid 4246 |
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