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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 4432 or weak linearity in ordsoexmid 4472) 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 2832 | . . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅}) | |
3 | velsn 3539 | . . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅) | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅) |
5 | noel 3362 | . . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | eleq2 2201 | . . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅)) | |
7 | 5, 6 | mtbiri 664 | . . . . . . . 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 609 | . . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
11 | 10 | gen2 1426 | . . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
12 | dftr2 4023 | . . . 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 3177 | . . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} | |
15 | ord0 4308 | . . . . 5 ⊢ Ord ∅ | |
16 | ordsucim 4411 | . . . . 5 ⊢ (Ord ∅ → Ord suc ∅) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ Ord suc ∅ |
18 | suc0 4328 | . . . . 5 ⊢ suc ∅ = {∅} | |
19 | ordeq 4289 | . . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅})) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅}) |
21 | 17, 20 | mpbi 144 | . . 3 ⊢ Ord {∅} |
22 | trssord 4297 | . . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑}) | |
23 | 13, 14, 21, 22 | mp3an 1315 | . 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑} |
24 | p0ex 4107 | . . . 4 ⊢ {∅} ∈ V | |
25 | 24 | rabex 4067 | . . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V |
26 | 25 | elon 4291 | . 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 1329 = wceq 1331 ∈ wcel 1480 {crab 2418 ⊆ wss 3066 ∅c0 3358 {csn 3522 Tr wtr 4021 Ord word 4279 Oncon0 4280 suc csuc 4282 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 |
This theorem is referenced by: ordtriexmid 4432 ordtri2orexmid 4433 ontr2exmid 4435 onsucsssucexmid 4437 ordsoexmid 4472 0elsucexmid 4475 ordpwsucexmid 4480 unfiexmid 6799 exmidonfinlem 7042 |
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