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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 4247 or weak linearity in ordsoexmid 4286) 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 102 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ 𝑧) | |
2 | elrabi 2695 | . . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 ∈ {∅}) | |
3 | velsn 3392 | . . . . . . . . 9 ⊢ (𝑧 ∈ {∅} ↔ 𝑧 = ∅) | |
4 | 2, 3 | sylib 127 | . . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑧 = ∅) |
5 | noel 3228 | . . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅ | |
6 | eleq2 2101 | . . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅)) | |
7 | 5, 6 | mtbiri 600 | . . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧) |
8 | 4, 7 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ 𝑦 ∈ 𝑧) |
9 | 8 | adantl 262 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → ¬ 𝑦 ∈ 𝑧) |
10 | 1, 9 | pm2.21dd 550 | . . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
11 | 10 | gen2 1339 | . . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) |
12 | dftr2 3856 | . . . 4 ⊢ (Tr {𝑥 ∈ {∅} ∣ 𝜑} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅} ∣ 𝜑})) | |
13 | 11, 12 | mpbir 134 | . . 3 ⊢ Tr {𝑥 ∈ {∅} ∣ 𝜑} |
14 | ssrab2 3025 | . . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} | |
15 | ord0 4128 | . . . . 5 ⊢ Ord ∅ | |
16 | ordsucim 4226 | . . . . 5 ⊢ (Ord ∅ → Ord suc ∅) | |
17 | 15, 16 | ax-mp 7 | . . . 4 ⊢ Ord suc ∅ |
18 | suc0 4148 | . . . . 5 ⊢ suc ∅ = {∅} | |
19 | ordeq 4109 | . . . . 5 ⊢ (suc ∅ = {∅} → (Ord suc ∅ ↔ Ord {∅})) | |
20 | 18, 19 | ax-mp 7 | . . . 4 ⊢ (Ord suc ∅ ↔ Ord {∅}) |
21 | 17, 20 | mpbi 133 | . . 3 ⊢ Ord {∅} |
22 | trssord 4117 | . . 3 ⊢ ((Tr {𝑥 ∈ {∅} ∣ 𝜑} ∧ {𝑥 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ Ord {∅}) → Ord {𝑥 ∈ {∅} ∣ 𝜑}) | |
23 | 13, 14, 21, 22 | mp3an 1232 | . 2 ⊢ Ord {𝑥 ∈ {∅} ∣ 𝜑} |
24 | p0ex 3939 | . . . 4 ⊢ {∅} ∈ V | |
25 | 24 | rabex 3901 | . . 3 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V |
26 | 25 | elon 4111 | . 2 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅} ∣ 𝜑}) |
27 | 23, 26 | mpbir 134 | 1 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 {crab 2310 ⊆ wss 2917 ∅c0 3224 {csn 3375 Tr wtr 3854 Ord word 4099 Oncon0 4100 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: ordtriexmid 4247 ordtri2orexmid 4248 ontr2exmid 4250 onsucsssucexmid 4252 ordsoexmid 4286 0elsucexmid 4289 ordpwsucexmid 4294 |
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