Theorem isinffi 9413

Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8723 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 9382	. . 3 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2		isinf 8723	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎))
3		breq2 5061	. . . . . 6 ⊢ (𝑎 = (card‘𝐵) → (𝑐 ≈ 𝑎 ↔ 𝑐 ≈ (card‘𝐵)))
4	3	anbi2d 630	. . . . 5 ⊢ (𝑎 = (card‘𝐵) → ((𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
5	4	exbidv 1916	. . . 4 ⊢ (𝑎 = (card‘𝐵) → (∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
6	5	rspcva 3619	. . 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 771	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9		ficardid 9383	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
10	9	ad2antlr 725	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11		entr 8553	. . . . . 6 ⊢ ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐 ≈ 𝐵)
12	8, 10, 11	syl2anc 586	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ 𝐵)
13	12	ensymd 8552	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝐵 ≈ 𝑐)
14		bren 8510	. . . 4 ⊢ (𝐵 ≈ 𝑐 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
15	13, 14	sylib 220	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
16		f1of1 6607	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝑐)
17		simplrl 775	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑐 ⊆ 𝐴)
18		f1ss 6573	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝑐 ∧ 𝑐 ⊆ 𝐴) → 𝑓:𝐵–1-1→𝐴)
19	16, 17, 18	syl2an2 684	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑓:𝐵–1-1→𝐴)
20	19	ex 415	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝐴))
21	20	eximdv 1912	. . 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 1930	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1531  ∃wex 1774   ∈ wcel 2108  ∀wral 3136   ⊆ wss 3934   class class class wbr 5057  –1-1→wf1 6345  –1-1-onto→wf1o 6347  ‘cfv 6348  ωcom 7572   ≈ cen 8498  Fincfn 8501  cardccrd 9356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360

This theorem is referenced by:  fidomtri  9414  hashdom  13732  erdsze2lem1  32443  eldioph2lem2  39348