Theorem isinffi 9979

Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9226 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 9948	. . 3 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2		isinf 9226	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎))
3		breq2 5114	. . . . . 6 ⊢ (𝑎 = (card‘𝐵) → (𝑐 ≈ 𝑎 ↔ 𝑐 ≈ (card‘𝐵)))
4	3	anbi2d 641	. . . . 5 ⊢ (𝑎 = (card‘𝐵) → ((𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
5	4	exbidv 1951	. . . 4 ⊢ (𝑎 = (card‘𝐵) → (∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
6	5	rspcva 3580	. . 3 ⊢ (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎)) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
7	1, 2, 6	syl2anr 608	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
8		simprr 784	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9		ficardid 9949	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
10	9	ad2antlr 739	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11		entr 9004	. . . . . 6 ⊢ ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐 ≈ 𝐵)
12	8, 10, 11	syl2anc 595	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ 𝐵)
13	12	ensymd 9003	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝐵 ≈ 𝑐)
14		bren 8954	. . . 4 ⊢ (𝐵 ≈ 𝑐 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
15	13, 14	sylib 221	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
16		f1of1 6821	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝑐)
17		simplrl 788	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑐 ⊆ 𝐴)
18		f1ss 6783	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝑐 ∧ 𝑐 ⊆ 𝐴) → 𝑓:𝐵–1-1→𝐴)
19	16, 17, 18	syl2an2 698	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑓:𝐵–1-1→𝐴)
20	19	ex 417	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝐴))
21	20	eximdv 1947	. . 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 16	. 2 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1→𝐴)
23	7, 22	exlimddv 1965	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906   class class class wbr 5110  –1-1→wf1 6535  –1-1-onto→wf1o 6537  ‘cfv 6538  ωcom 7863   ≈ cen 8941  Fincfn 8944  cardccrd 9922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7864  df-1o 8454  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-card 9926

This theorem is referenced by:  fidomtri  9980  hashdom  14417  erdsze2lem1  35676  eldioph2lem2  43475