Theorem elnn 4328

Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)

Assertion

Ref	Expression
elnn	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)

Proof of Theorem elnn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 2966	. . 3 ⊢ (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2		sseq1 2966	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3		sseq1 2966	. . 3 ⊢ (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4		sseq1 2966	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 ⊆ ω ↔ 𝐵 ⊆ ω))
5		0ss 3255	. . 3 ⊢ ∅ ⊆ ω
6		unss 3117	. . . . . 6 ⊢ ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7		vex 2560	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	snss 3494	. . . . . . 7 ⊢ (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
9	8	anbi2i 430	. . . . . 6 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10		df-suc 4108	. . . . . . 7 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
11	10	sseq1i 2969	. . . . . 6 ⊢ (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
12	6, 9, 11	3bitr4i 201	. . . . 5 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
13	12	biimpi 113	. . . 4 ⊢ ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
14	13	expcom 109	. . 3 ⊢ (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
15	1, 2, 3, 4, 5, 14	finds 4323	. 2 ⊢ (𝐵 ∈ ω → 𝐵 ⊆ ω)
16		ssel2 2940	. . 3 ⊢ ((𝐵 ⊆ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
17	16	ancoms 255	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ⊆ ω) → 𝐴 ∈ ω)
18	15, 17	sylan2 270	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  ∅c0 3224  {csn 3375  suc csuc 4102  ωcom 4313

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314

This theorem is referenced by:  ordom  4329  peano2b  4337  nndifsnid  6080  nnaordi  6081  nnmordi  6089  fidceq  6330  nnwetri  6354