Theorem elomssom 4709

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 4710. (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 3251	. 2 ⊢ (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2		sseq1 3251	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3		sseq1 3251	. 2 ⊢ (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4		sseq1 3251	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ ω ↔ 𝐴 ⊆ ω))
5		0ss 3535	. 2 ⊢ ∅ ⊆ ω
6		unss 3383	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7		vex 2806	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	snss 3813	. . . . . 6 ⊢ (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
9	8	anbi2i 457	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10		df-suc 4474	. . . . . 6 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
11	10	sseq1i 3254	. . . . 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 4704	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202   ∪ cun 3199   ⊆ wss 3201  ∅c0 3496  {csn 3673  suc csuc 4468  ωcom 4694

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-suc 4474  df-iom 4695

This theorem is referenced by:  elnn  4710  2ssom  6735  nninfwlpoimlemginf  7418  ennnfonelemg  13087