Theorem nndifsnid 6103

Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3531 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)

Assertion

Ref	Expression
nndifsnid	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Proof of Theorem nndifsnid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difsnss 3531	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
2	1	adantl 271	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
3		simpr 108	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
4		velsn 3415	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
5	3, 4	sylibr 132	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 ∈ {𝐵})
6		elun2 3140	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
7	5, 6	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
8		simplr 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
9		simpr 108	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
10	9, 4	sylnibr 634	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐵})
11	8, 10	eldifd 2983	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴 ∖ {𝐵}))
12		elun1 3139	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
13	11, 12	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
14		simpr 108	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
15		simpll 495	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ ω)
16		elnn 4346	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω)
17	14, 15, 16	syl2anc 403	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ω)
18		simplr 496	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴)
19		elnn 4346	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ ω)
20	18, 15, 19	syl2anc 403	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ω)
21		nndceq 6100	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝑥 = 𝐵)
22	17, 20, 21	syl2anc 403	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → DECID 𝑥 = 𝐵)
23		df-dc 776	. . . . . 6 ⊢ (DECID 𝑥 = 𝐵 ↔ (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
24	22, 23	sylib 120	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
25	7, 13, 24	mpjaodan 744	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
26	25	ex 113	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})))
27	26	ssrdv 3005	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
28	2, 27	eqssd 3016	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 661  DECID wdc 775   = wceq 1284   ∈ wcel 1433   ∖ cdif 2970   ∪ cun 2971   ⊆ wss 2973  {csn 3398  ωcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332

This theorem is referenced by:  phplem2  6339