Theorem isinffi 9916

Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9175 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 9885	. . 3 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2		isinf 9175	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎))
3		breq2 5089	. . . . . 6 ⊢ (𝑎 = (card‘𝐵) → (𝑐 ≈ 𝑎 ↔ 𝑐 ≈ (card‘𝐵)))
4	3	anbi2d 631	. . . . 5 ⊢ (𝑎 = (card‘𝐵) → ((𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
5	4	exbidv 1923	. . . 4 ⊢ (𝑎 = (card‘𝐵) → (∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
6	5	rspcva 3562	. . 3 ⊢ (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎)) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
7	1, 2, 6	syl2anr 598	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
8		simprr 773	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9		ficardid 9886	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
10	9	ad2antlr 728	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11		entr 8953	. . . . . 6 ⊢ ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐 ≈ 𝐵)
12	8, 10, 11	syl2anc 585	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ 𝐵)
13	12	ensymd 8952	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝐵 ≈ 𝑐)
14		bren 8903	. . . 4 ⊢ (𝐵 ≈ 𝑐 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
15	13, 14	sylib 218	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
16		f1of1 6779	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝑐)
17		simplrl 777	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑐 ⊆ 𝐴)
18		f1ss 6741	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝑐 ∧ 𝑐 ⊆ 𝐴) → 𝑓:𝐵–1-1→𝐴)
19	16, 17, 18	syl2an2 687	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑓:𝐵–1-1→𝐴)
20	19	ex 412	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝐴))
21	20	eximdv 1919	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (∃𝑓 𝑓:𝐵–1-1-onto→𝑐 → ∃𝑓 𝑓:𝐵–1-1→𝐴))
22	15, 21	mpd 15	. 2 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1→𝐴)
23	7, 22	exlimddv 1937	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889   class class class wbr 5085  –1-1→wf1 6495  –1-1-onto→wf1o 6497  ‘cfv 6498  ωcom 7817   ≈ cen 8890  Fincfn 8893  cardccrd 9859

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863

This theorem is referenced by:  fidomtri  9917  hashdom  14341  erdsze2lem1  35385  eldioph2lem2  43193