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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 4432 or weak linearity in ordsoexmid 4472) 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 3362 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | eleq2 2201 | . . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | mtbiri 664 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
4 | 0ex 4050 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snid 3551 | . . 3 ⊢ ∅ ∈ {∅} |
6 | biidd 171 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑)) | |
7 | 6 | elrab3 2836 | . . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
9 | 3, 8 | sylnib 665 | 1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {crab 2418 ∅c0 3358 {csn 3522 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-nul 4049 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-dif 3068 df-nul 3359 df-sn 3528 |
This theorem is referenced by: ordtriexmid 4432 ordtri2orexmid 4433 ontr2exmid 4435 onsucsssucexmid 4437 ordsoexmid 4472 0elsucexmid 4475 ordpwsucexmid 4480 |
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