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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0suc | GIF version |
Description: Proof of (biconditional form of) nn0suc 4615 from the core axioms of CZF. See also bj-nn0sucALT 14970. 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 14942 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥)) | |
2 | bj-omtrans 14948 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | |
3 | ssrexv 3232 | . . . . 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 789 | . . 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 4605 | . . . 4 ⊢ ∅ ∈ ω | |
8 | eleq1 2250 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω)) | |
9 | 7, 8 | mpbiri 168 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω) |
10 | bj-peano2 14931 | . . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω) | |
11 | eleq1a 2259 | . . . . . 6 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω)) | |
12 | 11 | imp 124 | . . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
13 | 10, 12 | sylan 283 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
14 | 13 | rexlimiva 2599 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω) |
15 | 9, 14 | jaoi 717 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
16 | 6, 15 | impbii 126 | 1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 709 = wceq 1363 ∈ wcel 2158 ∃wrex 2466 ⊆ wss 3141 ∅c0 3434 suc csuc 4377 ωcom 4601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-nul 4141 ax-pr 4221 ax-un 4445 ax-bd0 14805 ax-bdim 14806 ax-bdan 14807 ax-bdor 14808 ax-bdn 14809 ax-bdal 14810 ax-bdex 14811 ax-bdeq 14812 ax-bdel 14813 ax-bdsb 14814 ax-bdsep 14876 ax-infvn 14933 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-sn 3610 df-pr 3611 df-uni 3822 df-int 3857 df-suc 4383 df-iom 4602 df-bdc 14833 df-bj-ind 14919 |
This theorem is referenced by: bj-findis 14971 |
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