Theorem dif1en 6879

Description: If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)

Assertion

Ref	Expression
dif1en	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 998	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ≈ suc 𝑀)
2	1	ensymd 6783	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ≈ 𝐴)
3		bren 6747	. . 3 ⊢ (suc 𝑀 ≈ 𝐴 ↔ ∃𝑓 𝑓:suc 𝑀–1-1-onto→𝐴)
4	2, 3	sylib 122	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑓 𝑓:suc 𝑀–1-1-onto→𝐴)
5		peano2 4595	. . . . . . . 8 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
6		nnfi 6872	. . . . . . . 8 ⊢ (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
7	5, 6	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ Fin)
8	7	3ad2ant1 1018	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ∈ Fin)
9		enfii 6874	. . . . . 6 ⊢ ((suc 𝑀 ∈ Fin ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
10	8, 1, 9	syl2anc 411	. . . . 5 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ∈ Fin)
11	10	adantr 276	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝐴 ∈ Fin)
12		simpl3 1002	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑋 ∈ 𝐴)
13		f1of 5462	. . . . . 6 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓:suc 𝑀⟶𝐴)
14	13	adantl 277	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓:suc 𝑀⟶𝐴)
15		sucidg 4417	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
16	15	3ad2ant1 1018	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝑀 ∈ suc 𝑀)
17	16	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ suc 𝑀)
18	14, 17	ffvelcdmd 5653	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓‘𝑀) ∈ 𝐴)
19		fidifsnen 6870	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ (𝑓‘𝑀) ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
20	11, 12, 18, 19	syl3anc 1238	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
21		nnord 4612	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
22		orddif 4547	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
23	21, 22	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
24	23	3ad2ant1 1018	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝑀 = (suc 𝑀 ∖ {𝑀}))
25	24	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 = (suc 𝑀 ∖ {𝑀}))
26	23	eleq1d 2246	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → (𝑀 ∈ ω ↔ (suc 𝑀 ∖ {𝑀}) ∈ ω))
27	26	ibi 176	. . . . . . . 8 ⊢ (𝑀 ∈ ω → (suc 𝑀 ∖ {𝑀}) ∈ ω)
28	27	3ad2ant1 1018	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (suc 𝑀 ∖ {𝑀}) ∈ ω)
29	28	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (suc 𝑀 ∖ {𝑀}) ∈ ω)
30		dff1o2 5467	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 ↔ (𝑓 Fn suc 𝑀 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 𝐴))
31	30	simp2bi 1013	. . . . . . . 8 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → Fun ◡𝑓)
32	31	adantl 277	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → Fun ◡𝑓)
33		f1ofo 5469	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓:suc 𝑀–onto→𝐴)
34	33	adantl 277	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓:suc 𝑀–onto→𝐴)
35		f1orel 5465	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → Rel 𝑓)
36	35	adantl 277	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → Rel 𝑓)
37		resdm 4947	. . . . . . . . . . 11 ⊢ (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
38	36, 37	syl 14	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ dom 𝑓) = 𝑓)
39		f1odm 5466	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → dom 𝑓 = suc 𝑀)
40	39	reseq2d 4908	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → (𝑓 ↾ dom 𝑓) = (𝑓 ↾ suc 𝑀))
41	40	adantl 277	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ dom 𝑓) = (𝑓 ↾ suc 𝑀))
42	38, 41	eqtr3d 2212	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 = (𝑓 ↾ suc 𝑀))
43		foeq1 5435	. . . . . . . . 9 ⊢ (𝑓 = (𝑓 ↾ suc 𝑀) → (𝑓:suc 𝑀–onto→𝐴 ↔ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴))
44	42, 43	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓:suc 𝑀–onto→𝐴 ↔ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴))
45	34, 44	mpbid 147	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴)
46		simpl1 1000	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ ω)
47		f1osng 5503	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ω ∧ (𝑓‘𝑀) ∈ 𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)})
48	46, 18, 47	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)})
49		f1ofo 5469	. . . . . . . . 9 ⊢ ({⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)} → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)})
50	48, 49	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)})
51		f1ofn 5463	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓 Fn suc 𝑀)
52	51	adantl 277	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 Fn suc 𝑀)
53		fnressn 5703	. . . . . . . . . 10 ⊢ ((𝑓 Fn suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩})
54	52, 17, 53	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩})
55		foeq1 5435	. . . . . . . . 9 ⊢ ((𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩} → ((𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)} ↔ {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)}))
56	54, 55	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → ((𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)} ↔ {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)}))
57	50, 56	mpbird 167	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)})
58		resdif 5484	. . . . . . 7 ⊢ ((Fun ◡𝑓 ∧ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴 ∧ (𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)}) → (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)}))
59	32, 45, 57, 58	syl3anc 1238	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)}))
60		f1oeng 6757	. . . . . 6 ⊢ (((suc 𝑀 ∖ {𝑀}) ∈ ω ∧ (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)})) → (suc 𝑀 ∖ {𝑀}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
61	29, 59, 60	syl2anc 411	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (suc 𝑀 ∖ {𝑀}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
62	25, 61	eqbrtrd 4026	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
63	62	ensymd 6783	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀)
64		entr 6784	. . 3 ⊢ (((𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}) ∧ (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
65	20, 63, 64	syl2anc 411	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
66	4, 65	exlimddv 1898	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ∖ cdif 3127  {csn 3593  ⟨cop 3596   class class class wbr 4004  Ord word 4363  suc csuc 4366  ωcom 4590  ^◡ccnv 4626  dom cdm 4627  ran crn 4628   ↾ cres 4629  Rel wrel 4632  Fun wfun 5211   Fn wfn 5212  ⟶wf 5213  –onto→wfo 5215  –1-1-onto→wf1o 5216  ‘cfv 5217   ≈ cen 6738  Fincfn 6740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741  df-fin 6743

This theorem is referenced by:  dif1enen  6880  findcard  6888  findcard2  6889  findcard2s  6890  diffisn  6893  en2eleq  7194  en2other2  7195  zfz1isolem1  10820