Theorem elomssom 4653

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 4654. (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.

Step	Hyp	Ref	Expression
1		sseq1 3216	. 2 ⊢ (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2		sseq1 3216	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3		sseq1 3216	. 2 ⊢ (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4		sseq1 3216	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ ω ↔ 𝐴 ⊆ ω))
5		0ss 3499	. 2 ⊢ ∅ ⊆ ω
6		unss 3347	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7		vex 2775	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	snss 3768	. . . . . 6 ⊢ (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
9	8	anbi2i 457	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10		df-suc 4418	. . . . . 6 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
11	10	sseq1i 3219	. . . . 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 4648	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2176   ∪ cun 3164   ⊆ wss 3166  ∅c0 3460  {csn 3633  suc csuc 4412  ωcom 4638

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639

This theorem is referenced by:  elnn  4654  2ssom  6610  nninfwlpoimlemginf  7278  ennnfonelemg  12774