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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 4375 or weak linearity in ordsoexmid 4415) 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 3314 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | eleq2 2163 | . . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | mtbiri 641 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
4 | 0ex 3995 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snid 3503 | . . 3 ⊢ ∅ ∈ {∅} |
6 | biidd 171 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑)) | |
7 | 6 | elrab3 2794 | . . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
8 | 5, 7 | ax-mp 7 | . 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
9 | 3, 8 | sylnib 642 | 1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1299 ∈ wcel 1448 {crab 2379 ∅c0 3310 {csn 3474 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-nul 3994 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rab 2384 df-v 2643 df-dif 3023 df-nul 3311 df-sn 3480 |
This theorem is referenced by: ordtriexmid 4375 ordtri2orexmid 4376 ontr2exmid 4378 onsucsssucexmid 4380 ordsoexmid 4415 0elsucexmid 4418 ordpwsucexmid 4423 |
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