Theorem dif1enALT 9037

Description: Alternate proof of dif1en 8932 with fewer symbols using ax-pow 5286. (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
dif1enALT	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1enALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7727	. . . . 5 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2		breq2 5077	. . . . . . 7 ⊢ (𝑥 = suc 𝑀 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ suc 𝑀))
3	2	rspcev 3559	. . . . . 6 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8751	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
5	3, 4	sylibr 233	. . . . 5 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
6	1, 5	sylan 580	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7		diffi 8949	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8		isfi 8751	. . . . 5 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
9	7, 8	sylib 217	. . . 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 1131	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12		en2sn 8818	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ V) → {𝑋} ≈ {𝑥})
13	12	elvd 3436	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ≈ {𝑥})
14		nnord 7710	. . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥)
15		orddisj 6297	. . . . . . . 8 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17		incom 4134	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18		disjdif 4405	. . . . . . . . . 10 ⊢ ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
19	17, 18	eqtri 2766	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20		unen 8823	. . . . . . . . . 10 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
21	20	an4s 657	. . . . . . . . 9 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
22	19, 21	mpanl2 698	. . . . . . . 8 ⊢ (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
23	22	expcom 414	. . . . . . 7 ⊢ (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
24	13, 16, 23	syl2an 596	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
25	24	3ad2antl3 1186	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26		difsnid 4743	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27		df-suc 6265	. . . . . . . . . . 11 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
28	27	eqcomi 2747	. . . . . . . . . 10 ⊢ (𝑥 ∪ {𝑥}) = suc 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
30	26, 29	breq12d 5086	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31	30	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32	31	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
33		ensym 8776	. . . . . . . . . . 11 ⊢ (𝐴 ≈ suc 𝑀 → suc 𝑀 ≈ 𝐴)
34		entr 8779	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35		peano2 7727	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36		nneneq 8979	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
37	35, 36	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
38	34, 37	syl5ib 243	. . . . . . . . . . . 12 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
39	38	expd 416	. . . . . . . . . . 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 580	. . . . . . . . 9 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
42	41	imp 407	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
43	42	an32s 649	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
44	43	3adantl3 1167	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
45	32, 44	sylbid 239	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
46		peano4 7729	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 ↔ 𝑀 = 𝑥))
47	46	biimpd 228	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 → 𝑀 = 𝑥))
48	47	3ad2antl1 1184	. . . . 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 5077	. . . . 5 ⊢ (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
51	50	biimprcd 249	. . . 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 3211	. 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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3429   ∖ cdif 3883   ∪ cun 3884   ∩ cin 3885  ∅c0 4256  {csn 4561   class class class wbr 5073  Ord word 6258  suc csuc 6261  ωcom 7702   ≈ cen 8717  Fincfn 8720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-om 7703  df-1o 8284  df-er 8485  df-en 8721  df-fin 8724

This theorem is referenced by: (None)