Theorem nnsucuniel 6562

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 4547). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4568). (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 3455	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
2		uni0 3867	. . . . . . . 8 ⊢ ∪ ∅ = ∅
3	2	eleq2i 2263	. . . . . . 7 ⊢ (𝐴 ∈ ∪ ∅ ↔ 𝐴 ∈ ∅)
4	1, 3	mtbir 672	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∪ ∅
5		unieq 3849	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
6	5	eleq2d 2266	. . . . . 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 3849	. . . . . . . . . . . 12 ⊢ (𝐵 = suc 𝑛 → ∪ 𝐵 = ∪ suc 𝑛)
11	10	eleq2d 2266	. . . . . . . . . . 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 4649	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordtr 4414	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → Tr 𝑛)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Tr 𝑛)
18		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
19	18	unisuc 4449	. . . . . . . . . . . 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 2266	. . . . . . . . 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 6558	. . . . . . . . 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 2276	. . . . . 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 2615	. . . 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 4641	. . 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 4547	. 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 1364   ∈ wcel 2167  ∃wrex 2476  ∅c0 3451  ∪ cuni 3840  Tr wtr 4132  Ord word 4398  suc csuc 4401  ωcom 4627

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628

This theorem is referenced by: (None)