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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdeqsuc | GIF version |
Description: Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
bdeqsuc | ⊢ BOUNDED 𝑥 = suc 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcsuc 10000 | . . . 4 ⊢ BOUNDED suc 𝑦 | |
2 | 1 | bdss 9984 | . . 3 ⊢ BOUNDED 𝑥 ⊆ suc 𝑦 |
3 | bdcv 9968 | . . . . . . 7 ⊢ BOUNDED 𝑥 | |
4 | 3 | bdss 9984 | . . . . . 6 ⊢ BOUNDED 𝑦 ⊆ 𝑥 |
5 | 3 | bdsnss 9993 | . . . . . 6 ⊢ BOUNDED {𝑦} ⊆ 𝑥 |
6 | 4, 5 | ax-bdan 9935 | . . . . 5 ⊢ BOUNDED (𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) |
7 | unss 3117 | . . . . 5 ⊢ ((𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥) | |
8 | 6, 7 | bd0 9944 | . . . 4 ⊢ BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥 |
9 | df-suc 4108 | . . . . 5 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
10 | 9 | sseq1i 2969 | . . . 4 ⊢ (suc 𝑦 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥) |
11 | 8, 10 | bd0r 9945 | . . 3 ⊢ BOUNDED suc 𝑦 ⊆ 𝑥 |
12 | 2, 11 | ax-bdan 9935 | . 2 ⊢ BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥) |
13 | eqss 2960 | . 2 ⊢ (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥)) | |
14 | 12, 13 | bd0r 9945 | 1 ⊢ BOUNDED 𝑥 = suc 𝑦 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∪ cun 2915 ⊆ wss 2917 {csn 3375 suc csuc 4102 BOUNDED wbd 9932 |
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-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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-bd0 9933 ax-bdan 9935 ax-bdor 9936 ax-bdal 9938 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 |
This theorem depends on definitions: df-bi 110 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-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-suc 4108 df-bdc 9961 |
This theorem is referenced by: bj-bdsucel 10002 bj-nn0suc0 10075 |
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