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Theorem elomssom 4604
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.)
Assertion
Ref Expression
elomssom (𝐴 ∈ ω → 𝐴 ⊆ ω)

Proof of Theorem elomssom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef 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
87snss 3727 . . . . . 6 (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
98anbi2i 457 . . . . 5 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10 df-suc 4371 . . . . . 6 suc 𝑥 = (𝑥 ∪ {𝑥})
1110sseq1i 3181 . . . . 5 (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
126, 9, 113bitr4i 212 . . . 4 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
1312biimpi 120 . . 3 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
1413expcom 116 . 2 (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
151, 2, 3, 4, 5, 14finds 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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