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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0suc | GIF version |
Description: Proof of (biconditional form of) nn0suc 4327 from the core axioms of CZF. See also bj-nn0sucALT 10103. As a characterization of the elements of ω, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nn0suc | ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nn0suc0 10075 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥)) | |
2 | bj-omtrans 10081 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | |
3 | ssrexv 3005 | . . . . 5 ⊢ (𝐴 ⊆ ω → (∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
5 | 4 | orim2d 702 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))) |
6 | 1, 5 | mpd 13 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
7 | peano1 4317 | . . . 4 ⊢ ∅ ∈ ω | |
8 | eleq1 2100 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω)) | |
9 | 7, 8 | mpbiri 157 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω) |
10 | bj-peano2 10063 | . . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω) | |
11 | eleq1a 2109 | . . . . . 6 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω)) | |
12 | 11 | imp 115 | . . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
13 | 10, 12 | sylan 267 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
14 | 13 | rexlimiva 2428 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω) |
15 | 9, 14 | jaoi 636 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
16 | 6, 15 | impbii 117 | 1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ⊆ wss 2917 ∅c0 3224 suc csuc 4102 ωcom 4313 |
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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdim 9934 ax-bdan 9935 ax-bdor 9936 ax-bdn 9937 ax-bdal 9938 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 ax-infvn 10066 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 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-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bj-findis 10104 |
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