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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 4538 or weak linearity in ordsoexmid 4579) 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 3441 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | eleq2 2253 | . . 3 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | mtbiri 676 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
4 | 0ex 4145 | . . . 4 ⊢ ∅ ∈ V | |
5 | 4 | snid 3638 | . . 3 ⊢ ∅ ∈ {∅} |
6 | biidd 172 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑)) | |
7 | 6 | elrab3 2909 | . . 3 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
9 | 3, 8 | sylnib 677 | 1 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {crab 2472 ∅c0 3437 {csn 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-nul 4144 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-dif 3146 df-nul 3438 df-sn 3613 |
This theorem is referenced by: ordtriexmid 4538 ontriexmidim 4539 ordtri2orexmid 4540 ontr2exmid 4542 onsucsssucexmid 4544 ordsoexmid 4579 0elsucexmid 4582 ordpwsucexmid 4587 |
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