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Theorem isinffi 10023
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9291 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
isinffi ((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ βˆƒπ‘“ 𝑓:𝐡–1-1→𝐴)
Distinct variable groups:   𝐴,𝑓   𝐡,𝑓

Proof of Theorem isinffi
Dummy variables 𝑐 π‘Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ficardom 9992 . . 3 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
2 isinf 9291 . . 3 (Β¬ 𝐴 ∈ Fin β†’ βˆ€π‘Ž ∈ Ο‰ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž))
3 breq2 5156 . . . . . 6 (π‘Ž = (cardβ€˜π΅) β†’ (𝑐 β‰ˆ π‘Ž ↔ 𝑐 β‰ˆ (cardβ€˜π΅)))
43anbi2d 628 . . . . 5 (π‘Ž = (cardβ€˜π΅) β†’ ((𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž) ↔ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))))
54exbidv 1916 . . . 4 (π‘Ž = (cardβ€˜π΅) β†’ (βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž) ↔ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))))
65rspcva 3609 . . 3 (((cardβ€˜π΅) ∈ Ο‰ ∧ βˆ€π‘Ž ∈ Ο‰ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž)) β†’ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅)))
71, 2, 6syl2anr 595 . 2 ((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅)))
8 simprr 771 . . . . . 6 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝑐 β‰ˆ (cardβ€˜π΅))
9 ficardid 9993 . . . . . . 7 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
109ad2antlr 725 . . . . . 6 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
11 entr 9033 . . . . . 6 ((𝑐 β‰ˆ (cardβ€˜π΅) ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ 𝑐 β‰ˆ 𝐡)
128, 10, 11syl2anc 582 . . . . 5 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝑐 β‰ˆ 𝐡)
1312ensymd 9032 . . . 4 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝐡 β‰ˆ 𝑐)
14 bren 8980 . . . 4 (𝐡 β‰ˆ 𝑐 ↔ βˆƒπ‘“ 𝑓:𝐡–1-1-onto→𝑐)
1513, 14sylib 217 . . 3 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ βˆƒπ‘“ 𝑓:𝐡–1-1-onto→𝑐)
16 f1of1 6843 . . . . . 6 (𝑓:𝐡–1-1-onto→𝑐 β†’ 𝑓:𝐡–1-1→𝑐)
17 simplrl 775 . . . . . 6 ((((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) ∧ 𝑓:𝐡–1-1-onto→𝑐) β†’ 𝑐 βŠ† 𝐴)
18 f1ss 6804 . . . . . 6 ((𝑓:𝐡–1-1→𝑐 ∧ 𝑐 βŠ† 𝐴) β†’ 𝑓:𝐡–1-1→𝐴)
1916, 17, 18syl2an2 684 . . . . 5 ((((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) ∧ 𝑓:𝐡–1-1-onto→𝑐) β†’ 𝑓:𝐡–1-1→𝐴)
2019ex 411 . . . 4 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ (𝑓:𝐡–1-1-onto→𝑐 β†’ 𝑓:𝐡–1-1→𝐴))
2120eximdv 1912 . . 3 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ (βˆƒπ‘“ 𝑓:𝐡–1-1-onto→𝑐 β†’ βˆƒπ‘“ 𝑓:𝐡–1-1→𝐴))
2215, 21mpd 15 . 2 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ βˆƒπ‘“ 𝑓:𝐡–1-1→𝐴)
237, 22exlimddv 1930 1 ((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ βˆƒπ‘“ 𝑓:𝐡–1-1→𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3058   βŠ† wss 3949   class class class wbr 5152  β€“1-1β†’wf1 6550  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  Ο‰com 7876   β‰ˆ cen 8967  Fincfn 8970  cardccrd 9966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970
This theorem is referenced by:  fidomtri  10024  hashdom  14378  erdsze2lem1  34846  eldioph2lem2  42212
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