Theorem nnsucuniel 6454

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 4481). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4502). (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 3408	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
2		uni0 3810	. . . . . . . 8 ⊢ ∪ ∅ = ∅
3	2	eleq2i 2231	. . . . . . 7 ⊢ (𝐴 ∈ ∪ ∅ ↔ 𝐴 ∈ ∅)
4	1, 3	mtbir 661	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∪ ∅
5		unieq 3792	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
6	5	eleq2d 2234	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ ∅))
7	4, 6	mtbiri 665	. . . . 5 ⊢ (𝐵 = ∅ → ¬ 𝐴 ∈ ∪ 𝐵)
8	7	pm2.21d 609	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
9	8	adantl 275	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
10		unieq 3792	. . . . . . . . . . . 12 ⊢ (𝐵 = suc 𝑛 → ∪ 𝐵 = ∪ suc 𝑛)
11	10	eleq2d 2234	. . . . . . . . . . 11 ⊢ (𝐵 = suc 𝑛 → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ suc 𝑛))
12	11	ad2antll 483	. . . . . . . . . 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 525	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝑛 ∈ ω)
15		nnord 4583	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordtr 4350	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → Tr 𝑛)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Tr 𝑛)
18		vex 2724	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
19	18	unisuc 4385	. . . . . . . . . . . 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 2234	. . . . . . . . 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 6450	. . . . . . . . 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 526	. . . . . . 7 ⊢ (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 ∈ ∪ 𝐵) → 𝐵 = suc 𝑛)
28	26, 27	eleqtrrd 2244	. . . . . 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 2582	. . . 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 4575	. . 3 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
33	9, 31, 32	mpjaodan 788	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
34		sucunielr 4481	. 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 1342   ∈ wcel 2135  ∃wrex 2443  ∅c0 3404  ∪ cuni 3783  Tr wtr 4074  Ord word 4334  suc csuc 4337  ωcom 4561

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-uni 3784  df-int 3819  df-tr 4075  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562

This theorem is referenced by: (None)