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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 4247 or weak linearity in ordsoexmid 4286) 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 3228 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | eleq2 2101 | . . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | mtbiri 600 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
4 | 0ex 3884 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snid 3402 | . . 3 ⊢ ∅ ∈ {∅} |
6 | biidd 161 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑)) | |
7 | 6 | elrab3 2699 | . . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
8 | 5, 7 | ax-mp 7 | . 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
9 | 3, 8 | sylnib 601 | 1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 {crab 2310 ∅c0 3224 {csn 3375 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-dif 2920 df-nul 3225 df-sn 3381 |
This theorem is referenced by: ordtriexmid 4247 ordtri2orexmid 4248 ontr2exmid 4250 onsucsssucexmid 4252 ordsoexmid 4286 0elsucexmid 4289 ordpwsucexmid 4294 |
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