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Mirrors > Home > ILE Home > Th. List > elnn | GIF version |
Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
elnn | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 2966 | . . 3 ⊢ (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω)) | |
2 | sseq1 2966 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω)) | |
3 | sseq1 2966 | . . 3 ⊢ (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω)) | |
4 | sseq1 2966 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑦 ⊆ ω ↔ 𝐵 ⊆ ω)) | |
5 | 0ss 3255 | . . 3 ⊢ ∅ ⊆ ω | |
6 | unss 3117 | . . . . . 6 ⊢ ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω) | |
7 | vex 2560 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | snss 3494 | . . . . . . 7 ⊢ (𝑥 ∈ ω ↔ {𝑥} ⊆ ω) |
9 | 8 | anbi2i 430 | . . . . . 6 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω)) |
10 | df-suc 4108 | . . . . . . 7 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥}) | |
11 | 10 | sseq1i 2969 | . . . . . 6 ⊢ (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω) |
12 | 6, 9, 11 | 3bitr4i 201 | . . . . 5 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω) |
13 | 12 | biimpi 113 | . . . 4 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω) |
14 | 13 | expcom 109 | . . 3 ⊢ (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω)) |
15 | 1, 2, 3, 4, 5, 14 | finds 4323 | . 2 ⊢ (𝐵 ∈ ω → 𝐵 ⊆ ω) |
16 | ssel2 2940 | . . 3 ⊢ ((𝐵 ⊆ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω) | |
17 | 16 | ancoms 255 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ⊆ ω) → 𝐴 ∈ ω) |
18 | 15, 17 | sylan2 270 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∪ cun 2915 ⊆ wss 2917 ∅c0 3224 {csn 3375 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-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 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-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 |
This theorem is referenced by: ordom 4329 peano2b 4337 nndifsnid 6080 nnaordi 6081 nnmordi 6089 fidceq 6330 nnwetri 6354 |
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