Theorem dif1ennnALT 9291

Description: Alternate proof of dif1ennn 9175 using ax-pow 5359. (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 7888	. . . . 5 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2		breq2 5146	. . . . . . 7 ⊢ (𝑥 = suc 𝑀 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ suc 𝑀))
3	2	rspcev 3607	. . . . . 6 ⊢ ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8986	. . . . . 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 579	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7		diffi 9193	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8		isfi 8986	. . . . 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 1130	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12		en2sn 9055	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ V) → {𝑋} ≈ {𝑥})
13	12	elvd 3476	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ≈ {𝑥})
14		nnord 7870	. . . . . . . 8 ⊢ (𝑥 ∈ ω → Ord 𝑥)
15		orddisj 6401	. . . . . . . 8 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17		incom 4197	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18		disjdif 4467	. . . . . . . . . 10 ⊢ ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
19	17, 18	eqtri 2755	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20		unen 9060	. . . . . . . . . 10 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
21	20	an4s 659	. . . . . . . . 9 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
22	19, 21	mpanl2 700	. . . . . . . 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 595	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
25	24	3ad2antl3 1185	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26		difsnid 4809	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27		df-suc 6369	. . . . . . . . . . 11 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
28	27	eqcomi 2736	. . . . . . . . . 10 ⊢ (𝑥 ∪ {𝑥}) = suc 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
30	26, 29	breq12d 5155	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31	30	3ad2ant3 1133	. . . . . . 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 9013	. . . . . . . . . . 11 ⊢ (𝐴 ≈ suc 𝑀 → suc 𝑀 ≈ 𝐴)
34		entr 9016	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35		peano2 7888	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36		nneneq 9223	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
37	35, 36	sylan2 592	. . . . . . . . . . . . 13 ⊢ ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
38	34, 37	imbitrid 243	. . . . . . . . . . . 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 579	. . . . . . . . 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 651	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
44	43	3adantl3 1166	. . . . . 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 7890	. . . . . . 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 1183	. . . . 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 5146	. . . . 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 3150	. 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∃wrex 3065  Vcvv 3469   ∖ cdif 3941   ∪ cun 3942   ∩ cin 3943  ∅c0 4318  {csn 4624   class class class wbr 5142  Ord word 6362  suc csuc 6365  ωcom 7862   ≈ cen 8950  Fincfn 8953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7863  df-1o 8478  df-er 8716  df-en 8954  df-fin 8957

This theorem is referenced by: (None)