Theorem phplem4 8683

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.)

Hypotheses

Ref	Expression
phplem2.1	⊢ 𝐴 ∈ V
phplem2.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
phplem4	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Proof of Theorem phplem4

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8501	. 2 ⊢ (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵)
2		f1of1 6589	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵)
3	2	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝑓:suc 𝐴–1-1→suc 𝐵)
4		phplem2.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
5	4	sucex 7506	. . . . . . . . 9 ⊢ suc 𝐵 ∈ V
6		sssucid 6236	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
7		phplem2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
8		f1imaen2g 8553	. . . . . . . . . 10 ⊢ (((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ V) ∧ (𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ V)) → (𝑓 “ 𝐴) ≈ 𝐴)
9	6, 7, 8	mpanr12 704	. . . . . . . . 9 ⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ V) → (𝑓 “ 𝐴) ≈ 𝐴)
10	3, 5, 9	sylancl 589	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) ≈ 𝐴)
11	10	ensymd 8543	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓 “ 𝐴))
12		nnord 7568	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
13		orddif 6252	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
15	14	imaeq2d 5896	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16		f1ofn 6591	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓 Fn suc 𝐴)
17	7	sucid 6238	. . . . . . . . . . 11 ⊢ 𝐴 ∈ suc 𝐴
18		fnsnfv 6718	. . . . . . . . . . 11 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
19	16, 17, 18	sylancl 589	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → {(𝑓‘𝐴)} = (𝑓 “ {𝐴}))
20	19	difeq2d 4050	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
21		imadmrn 5906	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
22	21	eqcomi 2807	. . . . . . . . . . 11 ⊢ ran 𝑓 = (𝑓 “ dom 𝑓)
23		f1ofo 6597	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵)
24		forn 6568	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
26		f1odm 6594	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
27	26	imaeq2d 5896	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
28	22, 25, 27	3eqtr3a 2857	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
29	28	difeq1d 4049	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}))
30		dff1o3 6596	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓))
31		imadif 6408	. . . . . . . . . 10 ⊢ (Fun ◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
32	30, 31	simplbiim 508	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
33	20, 29, 32	3eqtr4rd 2844	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
34	15, 33	sylan9eq 2853	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)}))
35	11, 34	breqtrd 5056	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
36		fnfvelrn 6825	. . . . . . . . . 10 ⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓)
37	16, 17, 36	sylancl 589	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ ran 𝑓)
38	24	eleq2d 2875	. . . . . . . . . 10 ⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
39	23, 38	syl 17	. . . . . . . . 9 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵))
40	37, 39	mpbid 235	. . . . . . . 8 ⊢ (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → (𝑓‘𝐴) ∈ suc 𝐵)
41		fvex 6658	. . . . . . . . 9 ⊢ (𝑓‘𝐴) ∈ V
42	4, 41	phplem3 8682	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
43	40, 42	sylan2 595	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}))
44	43	ensymd 8543	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵)
45		entr 8544	. . . . . 6 ⊢ ((𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵)
46	35, 44, 45	syl2an 598	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵)) → 𝐴 ≈ 𝐵)
47	46	anandirs 678	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴–1-1-onto→suc 𝐵) → 𝐴 ≈ 𝐵)
48	47	ex 416	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
49	48	exlimdv 1934	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴–1-1-onto→suc 𝐵 → 𝐴 ≈ 𝐵))
50	1, 49	syl5bi 245	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  {csn 4525   class class class wbr 5030  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522  Ord word 6158  suc csuc 6161  Fun wfun 6318   Fn wfn 6319  –1-1→wf1 6321  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324  ωcom 7560   ≈ cen 8489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-er 8272  df-en 8493

This theorem is referenced by:  nneneq  8684