Theorem nnsucuniel 6496

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 4510). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4531). (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 3427	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
2		uni0 3837	. . . . . . . 8 ⊢ ∪ ∅ = ∅
3	2	eleq2i 2244	. . . . . . 7 ⊢ (𝐴 ∈ ∪ ∅ ↔ 𝐴 ∈ ∅)
4	1, 3	mtbir 671	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∪ ∅
5		unieq 3819	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
6	5	eleq2d 2247	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 ↔ 𝐴 ∈ ∪ ∅))
7	4, 6	mtbiri 675	. . . . 5 ⊢ (𝐵 = ∅ → ¬ 𝐴 ∈ ∪ 𝐵)
8	7	pm2.21d 619	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
9	8	adantl 277	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
10		unieq 3819	. . . . . . . . . . . 12 ⊢ (𝐵 = suc 𝑛 → ∪ 𝐵 = ∪ suc 𝑛)
11	10	eleq2d 2247	. . . . . . . . . . 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 4612	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordtr 4379	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → Tr 𝑛)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Tr 𝑛)
18		vex 2741	. . . . . . . . . . . . 13 ⊢ 𝑛 ∈ V
19	18	unisuc 4414	. . . . . . . . . . . 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 2247	. . . . . . . . 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 6492	. . . . . . . . 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 2257	. . . . . 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 2595	. . . 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 4604	. . 3 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
33	9, 31, 32	mpjaodan 798	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ∪ 𝐵 → suc 𝐴 ∈ 𝐵))
34		sucunielr 4510	. 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 1353   ∈ wcel 2148  ∃wrex 2456  ∅c0 3423  ∪ cuni 3810  Tr wtr 4102  Ord word 4363  suc csuc 4366  ωcom 4590

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591

This theorem is referenced by: (None)