Theorem phplem2 9203

Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5334. (Revised by BTernaryTau, 4-Nov-2024.)

Hypothesis

Ref	Expression
phplem2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
phplem2	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Proof of Theorem phplem2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8966	. 2 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2		f1of1 6820	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
3		nnfi 9166	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
4		sssucid 6444	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
5		f1imaenfi 9193	. . . . . . . . . 10 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ 𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ≈ 𝐴)
6	4, 5	mp3an2 1478	. . . . . . . . 9 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ 𝐴 ∈ Fin) → (𝑓 “ 𝐴) ≈ 𝐴)
7	2, 3, 6	syl2anr 609	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
8		ensymfib 9182	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
9	3, 8	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
10	9	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝐴 ≈ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐴) ≈ 𝐴))
11	7, 10	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
12		nnord 7874	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddif 6460	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
14	12, 13	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
15	14	imaeq2d 6060	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16		f1ofn 6822	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
17		phplem2.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
18	17	sucid 6446	. . . . . . . . . . 11 ⊢ 𝐴 ∈ suc 𝐴
19		fnsnfv 6961	. . . . . . . . . . 11 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
20	16, 18, 19	sylancl 598	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
21	20	difeq2d 4077	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22		imadmrn 6070	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
23	22	eqcomi 2771	. . . . . . . . . . 11 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
24		f1ofo 6829	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
25		forn 6796	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
26	24, 25	syl 18	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
27		f1odm 6825	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
28	27	imaeq2d 6060	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
29	23, 26, 28	3eqtr3a 2821	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
30	29	difeq1d 4076	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
31		dff1o3 6828	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
32		imadif 6621	. . . . . . . . . 10 ⊢ (Fun ◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
33	31, 32	simplbiim 514	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
34	21, 30, 33	3eqtr4rd 2808	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
35	15, 34	sylan9eq 2817	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
36	11, 35	breqtrd 5135	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
37		fnfvelrn 7077	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
38	16, 18, 37	sylancl 598	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ ran 𝑓)
39	25	eleq2d 2848	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
40	24, 39	syl 18	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
41	38, 40	mpbid 235	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ suc 𝐵)
42		phplem1 9202	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
43	41, 42	sylan2 605	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
44		nnfi 9166	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → 𝐵 ∈ Fin)
45		ensymfib 9182	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵))
46	44, 45	syl 18	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵))
47	46	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵))
48	43, 47	mpbid 235	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
49		entrfil 9183	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
50	3, 49	syl3an1 1181	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
51	48, 50	syl3an3 1183	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
52	51	3expa 1136	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)})) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
53	36, 52	syldanl 614	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
54	53	anandirs 692	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
55	54	ex 418	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
56	55	exlimdv 1966	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
57	1, 56	biimtrid 245	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  {csn 4587   class class class wbr 5107  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Ord word 6360  suc csuc 6363  Fun wfun 6531   Fn wfn 6532  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  ωcom 7866   ≈ cen 8953  Fincfn 8956

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7867  df-1o 8459  df-en 8957  df-fin 8960

This theorem is used by:  nneneq  9204