Theorem dif1enOLD 8885

Description: Obsolete version of dif1en 8818 as of 12-Aug-2024. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dif1enOLD	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1enOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7646	. . . . 5 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2		breq2 5043	. . . . . . 7 ⊢ (𝑥 = suc 𝑀 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ suc 𝑀))
3	2	rspcev 3527	. . . . . 6 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8630	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
5	3, 4	sylibr 237	. . . . 5 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
6	1, 5	sylan 583	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7		diffi 8884	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8		isfi 8630	. . . . 5 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
9	7, 8	sylib 221	. . . 4 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
10	6, 9	syl 17	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
11	10	3adant3 1134	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12		en2sn 8696	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ V) → {𝑋} ≈ {𝑥})
13	12	elvd 3405	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ≈ {𝑥})
14		nnord 7630	. . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥)
15		orddisj 6229	. . . . . . . 8 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17		incom 4101	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18		disjdif 4372	. . . . . . . . . 10 ⊢ ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
19	17, 18	eqtri 2759	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20		unen 8701	. . . . . . . . . 10 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
21	20	an4s 660	. . . . . . . . 9 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
22	19, 21	mpanl2 701	. . . . . . . 8 ⊢ (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
23	22	expcom 417	. . . . . . 7 ⊢ (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
24	13, 16, 23	syl2an 599	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
25	24	3ad2antl3 1189	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26		difsnid 4709	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27		df-suc 6197	. . . . . . . . . . 11 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
28	27	eqcomi 2745	. . . . . . . . . 10 ⊢ (𝑥 ∪ {𝑥}) = suc 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
30	26, 29	breq12d 5052	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31	30	3ad2ant3 1137	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32	31	adantr 484	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
33		ensym 8655	. . . . . . . . . . 11 ⊢ (𝐴 ≈ suc 𝑀 → suc 𝑀 ≈ 𝐴)
34		entr 8658	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35		peano2 7646	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36		nneneq 8807	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
37	35, 36	sylan2 596	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
38	34, 37	syl5ib 247	. . . . . . . . . . . 12 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
39	38	expd 419	. . . . . . . . . . 11 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ 𝐴 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
40	33, 39	syl5 34	. . . . . . . . . 10 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
41	1, 40	sylan 583	. . . . . . . . 9 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
42	41	imp 410	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
43	42	an32s 652	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
44	43	3adantl3 1170	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
45	32, 44	sylbid 243	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
46		peano4 7648	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 ↔ 𝑀 = 𝑥))
47	46	biimpd 232	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 → 𝑀 = 𝑥))
48	47	3ad2antl1 1187	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 → 𝑀 = 𝑥))
49	25, 45, 48	3syld 60	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → 𝑀 = 𝑥))
50		breq2 5043	. . . . 5 ⊢ (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
51	50	biimprcd 253	. . . 4 ⊢ ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝑀 = 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
52	49, 51	sylcom 30	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
53	52	rexlimdva 3193	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
54	11, 53	mpd 15	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∃wrex 3052  Vcvv 3398   ∖ cdif 3850   ∪ cun 3851   ∩ cin 3852  ∅c0 4223  {csn 4527   class class class wbr 5039  Ord word 6190  suc csuc 6193  ωcom 7622   ≈ cen 8601  Fincfn 8604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-1o 8180  df-er 8369  df-en 8605  df-fin 8608

This theorem is referenced by: (None)