Theorem nnsucuniel 6391

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 4426). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4446). (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 3367	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
2		uni0 3763	. . . . . . . 8 ⊢ ∪ ∅ = ∅
3	2	eleq2i 2206	. . . . . . 7 ⊢ (𝐴 ∈ ∪ ∅ ↔ 𝐴 ∈ ∅)
4	1, 3	mtbir 660	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∪ ∅
5		unieq 3745	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
6	5	eleq2d 2209	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ ∅))
7	4, 6	mtbiri 664	. . . . 5 ⊢ (𝐵 = ∅ → ¬ 𝐴 ∈ ∪ 𝐵)
8	7	pm2.21d 608	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
9	8	adantl 275	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
10		unieq 3745	. . . . . . . . . . . 12 ⊢ (𝐵 = suc 𝑛 → ∪ 𝐵 = ∪ suc 𝑛)
11	10	eleq2d 2209	. . . . . . . . . . 11 ⊢ (𝐵 = suc 𝑛 → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ suc 𝑛))
12	11	ad2antll 482	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ suc 𝑛))
13	12	biimpa 294	. . . . . . . . 9 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐴 ∈ ∪ suc 𝑛)
14		simplrl 524	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝑛 ∈ ω)
15		nnord 4525	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordtr 4300	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → Tr 𝑛)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Tr 𝑛)
18		vex 2689	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
19	18	unisuc 4335	. . . . . . . . . . . 12 ⊢ (Tr 𝑛 ↔ ∪ suc 𝑛 = 𝑛)
20	17, 19	sylib 121	. . . . . . . . . . 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 2209	. . . . . . . . 9 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → (𝐴 ∈ ∪ suc 𝑛 ↔ 𝐴 ∈ 𝑛))
23	13, 22	mpbid 146	. . . . . . . 8 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐴 ∈ 𝑛)
24		nnsucelsuc 6387	. . . . . . . . 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 146	. . . . . . 7 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → suc 𝐴 ∈ suc 𝑛)
27		simplrr 525	. . . . . . 7 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐵 = suc 𝑛)
28	26, 27	eleqtrrd 2219	. . . . . 6 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → suc 𝐴 ∈ 𝐵)
29	28	ex 114	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
30	29	rexlimdvaa 2550	. . . 4 ⊢ (𝐵 ∈ ω → (∃𝑛 ∈ ω 𝐵 = suc 𝑛 → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵)))
31	30	imp 123	. . 3 ⊢ ((𝐵 ∈ ω ∧ ∃𝑛 ∈ ω 𝐵 = suc 𝑛) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
32		nn0suc 4518	. . 3 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
33	9, 31, 32	mpjaodan 787	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
34		sucunielr 4426	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵)
35	33, 34	impbid1 141	1 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  ∅c0 3363  ∪ cuni 3736  Tr wtr 4026  Ord word 4284  suc csuc 4287  ωcom 4504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505

This theorem is referenced by: (None)