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Mirrors > Home > ILE Home > Th. List > 2ordpr | GIF version |
Description: Version of 2on 6330 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Ref | Expression |
---|---|
2ordpr | ⊢ Ord {∅, {∅}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ord0 4321 | . . 3 ⊢ Ord ∅ | |
2 | ordsucim 4424 | . . 3 ⊢ (Ord ∅ → Ord suc ∅) | |
3 | ordsucim 4424 | . . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅) | |
4 | 1, 2, 3 | mp2b 8 | . 2 ⊢ Ord suc suc ∅ |
5 | df-suc 4301 | . . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}}) | |
6 | suc0 4341 | . . . . 5 ⊢ suc ∅ = {∅} | |
7 | suceq 4332 | . . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅}) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ suc suc ∅ = suc {∅} |
9 | df-pr 3539 | . . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}}) | |
10 | 5, 8, 9 | 3eqtr4i 2171 | . . 3 ⊢ suc suc ∅ = {∅, {∅}} |
11 | ordeq 4302 | . . 3 ⊢ (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}})) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (Ord suc suc ∅ ↔ Ord {∅, {∅}}) |
13 | 4, 12 | mpbi 144 | 1 ⊢ Ord {∅, {∅}} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∪ cun 3074 ∅c0 3368 {csn 3532 {cpr 3533 Ord word 4292 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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 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-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-tr 4035 df-iord 4296 df-suc 4301 |
This theorem is referenced by: ontr2exmid 4448 ordtri2or2exmidlem 4449 onsucelsucexmidlem 4452 |
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