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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0suc | GIF version |
Description: Proof of (biconditional form of) nn0suc 4588 from the core axioms of CZF. See also bj-nn0sucALT 14013. 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 13985 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥)) | |
2 | bj-omtrans 13991 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | |
3 | ssrexv 3212 | . . . . 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 783 | . . 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 4578 | . . . 4 ⊢ ∅ ∈ ω | |
8 | eleq1 2233 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω)) | |
9 | 7, 8 | mpbiri 167 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω) |
10 | bj-peano2 13974 | . . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω) | |
11 | eleq1a 2242 | . . . . . 6 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω)) | |
12 | 11 | imp 123 | . . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
13 | 10, 12 | sylan 281 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
14 | 13 | rexlimiva 2582 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω) |
15 | 9, 14 | jaoi 711 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω) |
16 | 6, 15 | impbii 125 | 1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 ⊆ wss 3121 ∅c0 3414 suc csuc 4350 ωcom 4574 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-nul 4115 ax-pr 4194 ax-un 4418 ax-bd0 13848 ax-bdim 13849 ax-bdan 13850 ax-bdor 13851 ax-bdn 13852 ax-bdal 13853 ax-bdex 13854 ax-bdeq 13855 ax-bdel 13856 ax-bdsb 13857 ax-bdsep 13919 ax-infvn 13976 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-iom 4575 df-bdc 13876 df-bj-ind 13962 |
This theorem is referenced by: bj-findis 14014 |
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