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Theorem elomssom 4562
 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 4563. (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 3151 . 2 (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2 sseq1 3151 . 2 (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3 sseq1 3151 . 2 (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4 sseq1 3151 . 2 (𝑦 = 𝐴 → (𝑦 ⊆ ω ↔ 𝐴 ⊆ ω))
5 0ss 3432 . 2 ∅ ⊆ ω
6 unss 3281 . . . . 5 ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7 vex 2715 . . . . . . 7 𝑥 ∈ V
87snss 3685 . . . . . 6 (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
98anbi2i 453 . . . . 5 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10 df-suc 4330 . . . . . 6 suc 𝑥 = (𝑥 ∪ {𝑥})
1110sseq1i 3154 . . . . 5 (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
126, 9, 113bitr4i 211 . . . 4 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
1312biimpi 119 . . 3 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
1413expcom 115 . 2 (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
151, 2, 3, 4, 5, 14finds 4557 1 (𝐴 ∈ ω → 𝐴 ⊆ ω)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2128   ∪ cun 3100   ⊆ wss 3102  ∅c0 3394  {csn 3560  suc csuc 4324  ωcom 4547 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-iinf 4545 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4330  df-iom 4548 This theorem is referenced by:  elnn  4563  2ssom  13337
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