Theorem dif1ennnALT 9281

Description: Alternate proof of dif1ennn 9173 using ax-pow 5335. (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
dif1ennnALT	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1ennnALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7884	. . . . 5 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2		breq2 5123	. . . . . . 7 ⊢ (𝑥 = suc 𝑀 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ suc 𝑀))
3	2	rspcev 3601	. . . . . 6 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8988	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
5	3, 4	sylibr 234	. . . . 5 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
6	1, 5	sylan 580	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7		diffi 9187	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8		isfi 8988	. . . . 5 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
9	7, 8	sylib 218	. . . 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 1132	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12		en2sn 9053	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ V) → {𝑋} ≈ {𝑥})
13	12	elvd 3465	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ≈ {𝑥})
14		nnord 7867	. . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥)
15		orddisj 6390	. . . . . . . 8 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17		incom 4184	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18		disjdif 4447	. . . . . . . . . 10 ⊢ ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
19	17, 18	eqtri 2758	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20		unen 9058	. . . . . . . . . 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 413	. . . . . . 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 1188	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26		difsnid 4786	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27		df-suc 6358	. . . . . . . . . . 11 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
28	27	eqcomi 2744	. . . . . . . . . 10 ⊢ (𝑥 ∪ {𝑥}) = suc 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
30	26, 29	breq12d 5132	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31	30	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32	31	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
33		ensym 9015	. . . . . . . . . . 11 ⊢ (𝐴 ≈ suc 𝑀 → suc 𝑀 ≈ 𝐴)
34		entr 9018	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35		peano2 7884	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36		nneneq 9218	. . . . . . . . . . . . . 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	imbitrid 244	. . . . . . . . . . . 12 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
39	38	expd 415	. . . . . . . . . . 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 406	. . . . . . . 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 1169	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
45	32, 44	sylbid 240	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
46		peano4 7886	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 ↔ 𝑀 = 𝑥))
47	46	biimpd 229	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 → 𝑀 = 𝑥))
48	47	3ad2antl1 1186	. . . . 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 5123	. . . . 5 ⊢ (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
51	50	biimprcd 250	. . . 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 3141	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925  ∅c0 4308  {csn 4601   class class class wbr 5119  Ord word 6351  suc csuc 6354  ωcom 7859   ≈ cen 8954  Fincfn 8957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-om 7860  df-1o 8478  df-er 8717  df-en 8958  df-fin 8961

This theorem is referenced by: (None)