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Theorem isinffi 9986
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9259 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 9955 . . 3 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
2 isinf 9259 . . 3 (Β¬ 𝐴 ∈ Fin β†’ βˆ€π‘Ž ∈ Ο‰ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž))
3 breq2 5145 . . . . . 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 3604 . . 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 596 . 2 ((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅)))
8 simprr 770 . . . . . 6 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝑐 β‰ˆ (cardβ€˜π΅))
9 ficardid 9956 . . . . . . 7 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
109ad2antlr 724 . . . . . 6 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
11 entr 9001 . . . . . 6 ((𝑐 β‰ˆ (cardβ€˜π΅) ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ 𝑐 β‰ˆ 𝐡)
128, 10, 11syl2anc 583 . . . . 5 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝑐 β‰ˆ 𝐡)
1312ensymd 9000 . . . 4 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝐡 β‰ˆ 𝑐)
14 bren 8948 . . . 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 6825 . . . . . 6 (𝑓:𝐡–1-1-onto→𝑐 β†’ 𝑓:𝐡–1-1→𝑐)
17 simplrl 774 . . . . . 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 6786 . . . . . 6 ((𝑓:𝐡–1-1→𝑐 ∧ 𝑐 βŠ† 𝐴) β†’ 𝑓:𝐡–1-1→𝐴)
1916, 17, 18syl2an2 683 . . . . 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 412 . . . 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 395   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943   class class class wbr 5141  β€“1-1β†’wf1 6533  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  Ο‰com 7851   β‰ˆ cen 8935  Fincfn 8938  cardccrd 9929
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7852  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933
This theorem is referenced by:  fidomtri  9987  hashdom  14341  erdsze2lem1  34721  eldioph2lem2  42059
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