Theorem nnsucuniel 6571

Description: Given an element 𝐴 of the union of a natural number 𝐵, suc 𝐴 is an element of 𝐵 itself. The reverse direction holds for all ordinals (sucunielr 4556). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4577). (Contributed by Jim Kingdon, 13-Mar-2022.)

Assertion

Ref	Expression
nnsucuniel	⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))

Proof of Theorem nnsucuniel

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3463	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
2		uni0 3876	. . . . . . . 8 ⊢ ∪ ∅ = ∅
3	2	eleq2i 2271	. . . . . . 7 ⊢ (𝐴 ∈ ∪ ∅ ↔ 𝐴 ∈ ∅)
4	1, 3	mtbir 672	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∪ ∅
5		unieq 3858	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
6	5	eleq2d 2274	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ ∅))
7	4, 6	mtbiri 676	. . . . 5 ⊢ (𝐵 = ∅ → ¬ 𝐴 ∈ ∪ 𝐵)
8	7	pm2.21d 620	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
9	8	adantl 277	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
10		unieq 3858	. . . . . . . . . . . 12 ⊢ (𝐵 = suc 𝑛 → ∪ 𝐵 = ∪ suc 𝑛)
11	10	eleq2d 2274	. . . . . . . . . . 11 ⊢ (𝐵 = suc 𝑛 → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ suc 𝑛))
12	11	ad2antll 491	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ suc 𝑛))
13	12	biimpa 296	. . . . . . . . 9 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐴 ∈ ∪ suc 𝑛)
14		simplrl 535	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝑛 ∈ ω)
15		nnord 4658	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordtr 4423	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → Tr 𝑛)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Tr 𝑛)
18		vex 2774	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
19	18	unisuc 4458	. . . . . . . . . . . 12 ⊢ (Tr 𝑛 ↔ ∪ suc 𝑛 = 𝑛)
20	17, 19	sylib 122	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ∪ suc 𝑛 = 𝑛)
21	14, 20	syl 14	. . . . . . . . . 10 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → ∪ suc 𝑛 = 𝑛)
22	21	eleq2d 2274	. . . . . . . . 9 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → (𝐴 ∈ ∪ suc 𝑛 ↔ 𝐴 ∈ 𝑛))
23	13, 22	mpbid 147	. . . . . . . 8 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐴 ∈ 𝑛)
24		nnsucelsuc 6567	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (𝐴 ∈ 𝑛 ↔ suc 𝐴 ∈ suc 𝑛))
25	14, 24	syl 14	. . . . . . . 8 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → (𝐴 ∈ 𝑛 ↔ suc 𝐴 ∈ suc 𝑛))
26	23, 25	mpbid 147	. . . . . . 7 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → suc 𝐴 ∈ suc 𝑛)
27		simplrr 536	. . . . . . 7 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐵 = suc 𝑛)
28	26, 27	eleqtrrd 2284	. . . . . 6 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → suc 𝐴 ∈ 𝐵)
29	28	ex 115	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
30	29	rexlimdvaa 2623	. . . 4 ⊢ (𝐵 ∈ ω → (∃𝑛 ∈ ω 𝐵 = suc 𝑛 → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵)))
31	30	imp 124	. . 3 ⊢ ((𝐵 ∈ ω ∧ ∃𝑛 ∈ ω 𝐵 = suc 𝑛) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
32		nn0suc 4650	. . 3 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
33	9, 31, 32	mpjaodan 799	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
34		sucunielr 4556	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵)
35	33, 34	impbid1 142	1 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  ∃wrex 2484  ∅c0 3459  ∪ cuni 3849  Tr wtr 4141  Ord word 4407  suc csuc 4410  ωcom 4636

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-tr 4142  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637

This theorem is referenced by: (None)