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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 4503 or weak linearity in ordsoexmid 4544) 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 3418 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | eleq2 2234 | . . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | mtbiri 670 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
4 | 0ex 4114 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snid 3612 | . . 3 ⊢ ∅ ∈ {∅} |
6 | biidd 171 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑)) | |
7 | 6 | elrab3 2887 | . . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
9 | 3, 8 | sylnib 671 | 1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1348 ∈ wcel 2141 {crab 2452 ∅c0 3414 {csn 3581 |
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-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rab 2457 df-v 2732 df-dif 3123 df-nul 3415 df-sn 3587 |
This theorem is referenced by: ordtriexmid 4503 ontriexmidim 4504 ordtri2orexmid 4505 ontr2exmid 4507 onsucsssucexmid 4509 ordsoexmid 4544 0elsucexmid 4547 ordpwsucexmid 4552 |
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