Theorem isinffi 8818

Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8173 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
isinffi	⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem isinffi

Dummy variables 𝑐 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ficardom 8787	. . 3 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2		isinf 8173	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎))
3		breq2 4657	. . . . . 6 ⊢ (𝑎 = (card‘𝐵) → (𝑐 ≈ 𝑎 ↔ 𝑐 ≈ (card‘𝐵)))
4	3	anbi2d 740	. . . . 5 ⊢ (𝑎 = (card‘𝐵) → ((𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
5	4	exbidv 1850	. . . 4 ⊢ (𝑎 = (card‘𝐵) → (∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
6	5	rspcva 3307	. . 3 ⊢ (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎)) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
7	1, 2, 6	syl2anr 495	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
8		simprr 796	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9		ficardid 8788	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
10	9	ad2antlr 763	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11		entr 8008	. . . . . 6 ⊢ ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐 ≈ 𝐵)
12	8, 10, 11	syl2anc 693	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ 𝐵)
13	12	ensymd 8007	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝐵 ≈ 𝑐)
14		bren 7964	. . . 4 ⊢ (𝐵 ≈ 𝑐 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
15	13, 14	sylib 208	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
16		f1of1 6136	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝑐)
17	16	adantl 482	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑓:𝐵–1-1→𝑐)
18		simplrl 800	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑐 ⊆ 𝐴)
19		f1ss 6106	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝑐 ∧ 𝑐 ⊆ 𝐴) → 𝑓:𝐵–1-1→𝐴)
20	17, 18, 19	syl2anc 693	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑓:𝐵–1-1→𝐴)
21	20	ex 450	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝐴))
22	21	eximdv 1846	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (∃𝑓 𝑓:𝐵–1-1-onto→𝑐 → ∃𝑓 𝑓:𝐵–1-1→𝐴))
23	15, 22	mpd 15	. 2 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1→𝐴)
24	7, 23	exlimddv 1863	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574   class class class wbr 4653  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  ωcom 7065   ≈ cen 7952  Fincfn 7955  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fidomtri  8819  hashdom  13168  erdsze2lem1  31185  eldioph2lem2  37324