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Mirrors > Home > ILE Home > Th. List > elomssom | GIF version |
Description: A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4605. (Revised by BJ, 7-Aug-2024.) |
Ref | Expression |
---|---|
elomssom | ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3178 | . 2 ⊢ (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω)) | |
2 | sseq1 3178 | . 2 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω)) | |
3 | sseq1 3178 | . 2 ⊢ (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω)) | |
4 | sseq1 3178 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ ω ↔ 𝐴 ⊆ ω)) | |
5 | 0ss 3461 | . 2 ⊢ ∅ ⊆ ω | |
6 | unss 3309 | . . . . 5 ⊢ ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω) | |
7 | vex 2740 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 7 | snss 3727 | . . . . . 6 ⊢ (𝑥 ∈ ω ↔ {𝑥} ⊆ ω) |
9 | 8 | anbi2i 457 | . . . . 5 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω)) |
10 | df-suc 4371 | . . . . . 6 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥}) | |
11 | 10 | sseq1i 3181 | . . . . 5 ⊢ (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω) |
12 | 6, 9, 11 | 3bitr4i 212 | . . . 4 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω) |
13 | 12 | biimpi 120 | . . 3 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω) |
14 | 13 | expcom 116 | . 2 ⊢ (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω)) |
15 | 1, 2, 3, 4, 5, 14 | finds 4599 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ∪ cun 3127 ⊆ wss 3129 ∅c0 3422 {csn 3592 suc csuc 4365 ωcom 4589 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4371 df-iom 4590 |
This theorem is referenced by: elnn 4605 2ssom 6524 nninfwlpoimlemginf 7173 ennnfonelemg 12398 |
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