Theorem elomssom 4696

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 4697. (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 3247	. 2 ⊢ (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2		sseq1 3247	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3		sseq1 3247	. 2 ⊢ (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4		sseq1 3247	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ ω ↔ 𝐴 ⊆ ω))
5		0ss 3530	. 2 ⊢ ∅ ⊆ ω
6		unss 3378	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7		vex 2802	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	snss 3802	. . . . . 6 ⊢ (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
9	8	anbi2i 457	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10		df-suc 4461	. . . . . 6 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
11	10	sseq1i 3250	. . . . 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 4691	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200   ∪ cun 3195   ⊆ wss 3197  ∅c0 3491  {csn 3666  suc csuc 4455  ωcom 4681

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-suc 4461  df-iom 4682

This theorem is referenced by:  elnn  4697  2ssom  6668  nninfwlpoimlemginf  7339  ennnfonelemg  12969