Theorem dif1ennnALT 9306

Description: Alternate proof of dif1ennn 9216 using ax-pow 5334. (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 7935	. . . . 5 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2		breq2 5112	. . . . . . 7 ⊢ (𝑥 = suc 𝑀 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ suc 𝑀))
3	2	rspcev 3580	. . . . . 6 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 9040	. . . . . 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 592	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7		diffi 9228	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8		isfi 9040	. . . . 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 18	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
11	10	3adant3 1150	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12		en2sn 9107	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ V) → {𝑋} ≈ {𝑥})
13	12	elvd 3460	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ≈ {𝑥})
14		nnord 7919	. . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥)
15		orddisj 6427	. . . . . . . 8 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
16	14, 15	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17		incom 4109	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18		disjdif 4429	. . . . . . . . . 10 ⊢ ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
19	17, 18	eqtri 2786	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20		unen 9111	. . . . . . . . . 10 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
21	20	an4s 673	. . . . . . . . 9 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
22	19, 21	mpanl2 714	. . . . . . . 8 ⊢ (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
23	22	expcom 419	. . . . . . 7 ⊢ (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
24	13, 16, 23	syl2an 608	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
25	24	3ad2antl3 1206	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26		difsnid 4773	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27		df-suc 6394	. . . . . . . . . . 11 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
28	27	eqcomi 2772	. . . . . . . . . 10 ⊢ (𝑥 ∪ {𝑥}) = suc 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
30	26, 29	breq12d 5121	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31	30	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32	31	adantr 486	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
33		ensym 9068	. . . . . . . . . . 11 ⊢ (𝐴 ≈ suc 𝑀 → suc 𝑀 ≈ 𝐴)
34		entr 9071	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35		peano2 7935	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36		nneneq 9259	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
37	35, 36	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
38	34, 37	imbitrid 247	. . . . . . . . . . . 12 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
39	38	expd 421	. . . . . . . . . . 11 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ 𝐴 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
40	33, 39	syl5 35	. . . . . . . . . 10 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
41	1, 40	sylan 592	. . . . . . . . 9 ⊢ ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
42	41	imp 412	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
43	42	an32s 665	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
44	43	3adantl3 1187	. . . . . 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 7938	. . . . . . 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 1204	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥 → 𝑀 = 𝑥))
49	25, 45, 48	3syld 61	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → 𝑀 = 𝑥))
50		breq2 5112	. . . . 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 31	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
53	52	rexlimdva 3167	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
54	11, 53	mpd 16	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588   class class class wbr 5108  Ord word 6387  suc csuc 6390  ωcom 7911   ≈ cen 9008  Fincfn 9011

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-1o 8514  df-er 8755  df-en 9012  df-fin 9015

This theorem is used by: (None)