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Mirrors > Home > ILE Home > Th. List > ordtriexmidlem | 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 4445 or weak linearity in ordsoexmid 4485) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Ref | Expression |
---|---|
ordtriexmidlem | ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ 𝑧) | |
2 | elrabi 2841 | . . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅}) | |
3 | velsn 3549 | . . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅) | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅) |
5 | noel 3372 | . . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | eleq2 2204 | . . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅)) | |
7 | 5, 6 | mtbiri 665 | . . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧) |
8 | 4, 7 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦 ∈ 𝑧) |
9 | 8 | adantl 275 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦 ∈ 𝑧) |
10 | 1, 9 | pm2.21dd 610 | . . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
11 | 10 | gen2 1427 | . . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
12 | dftr2 4036 | . . . 4 ⊢ (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})) | |
13 | 11, 12 | mpbir 145 | . . 3 ⊢ Tr {𝑥 ∈ {∅} ∣ 𝜑} |
14 | ssrab2 3187 | . . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} | |
15 | ord0 4321 | . . . . 5 ⊢ Ord ∅ | |
16 | ordsucim 4424 | . . . . 5 ⊢ (Ord ∅ → Ord suc ∅) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ Ord suc ∅ |
18 | suc0 4341 | . . . . 5 ⊢ suc ∅ = {∅} | |
19 | ordeq 4302 | . . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅})) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅}) |
21 | 17, 20 | mpbi 144 | . . 3 ⊢ Ord {∅} |
22 | trssord 4310 | . . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑}) | |
23 | 13, 14, 21, 22 | mp3an 1316 | . 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑} |
24 | p0ex 4120 | . . . 4 ⊢ {∅} ∈ V | |
25 | 24 | rabex 4080 | . . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V |
26 | 25 | elon 4304 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑}) |
27 | 23, 26 | mpbir 145 | 1 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1330 = wceq 1332 ∈ wcel 1481 {crab 2421 ⊆ wss 3076 ∅c0 3368 {csn 3532 Tr wtr 4034 Ord word 4292 Oncon0 4293 suc csuc 4295 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 |
This theorem is referenced by: ordtriexmid 4445 ordtri2orexmid 4446 ontr2exmid 4448 onsucsssucexmid 4450 ordsoexmid 4485 0elsucexmid 4488 ordpwsucexmid 4493 unfiexmid 6814 exmidonfinlem 7066 |
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