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Mirrors > Home > ILE Home > Th. List > 2ordpr | GIF version |
Description: Version of 2on 6322 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 4313 | . . 3 ⊢ Ord ∅ | |
2 | ordsucim 4416 | . . 3 ⊢ (Ord ∅ → Ord suc ∅) | |
3 | ordsucim 4416 | . . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅) | |
4 | 1, 2, 3 | mp2b 8 | . 2 ⊢ Ord suc suc ∅ |
5 | df-suc 4293 | . . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}}) | |
6 | suc0 4333 | . . . . 5 ⊢ suc ∅ = {∅} | |
7 | suceq 4324 | . . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅}) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ suc suc ∅ = suc {∅} |
9 | df-pr 3534 | . . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}}) | |
10 | 5, 8, 9 | 3eqtr4i 2170 | . . 3 ⊢ suc suc ∅ = {∅, {∅}} |
11 | ordeq 4294 | . . 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 1331 ∪ cun 3069 ∅c0 3363 {csn 3527 {cpr 3528 Ord word 4284 suc csuc 4287 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-suc 4293 |
This theorem is referenced by: ontr2exmid 4440 ordtri2or2exmidlem 4441 onsucelsucexmidlem 4444 |
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