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Theorem isinffi 9933
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9207 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 9902 . . 3 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
2 isinf 9207 . . 3 (Β¬ 𝐴 ∈ Fin β†’ βˆ€π‘Ž ∈ Ο‰ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž))
3 breq2 5110 . . . . . 6 (π‘Ž = (cardβ€˜π΅) β†’ (𝑐 β‰ˆ π‘Ž ↔ 𝑐 β‰ˆ (cardβ€˜π΅)))
43anbi2d 630 . . . . 5 (π‘Ž = (cardβ€˜π΅) β†’ ((𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ π‘Ž) ↔ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))))
54exbidv 1925 . . . 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 3578 . . 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 598 . 2 ((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ βˆƒπ‘(𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅)))
8 simprr 772 . . . . . 6 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝑐 β‰ˆ (cardβ€˜π΅))
9 ficardid 9903 . . . . . . 7 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
109ad2antlr 726 . . . . . 6 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
11 entr 8949 . . . . . 6 ((𝑐 β‰ˆ (cardβ€˜π΅) ∧ (cardβ€˜π΅) β‰ˆ 𝐡) β†’ 𝑐 β‰ˆ 𝐡)
128, 10, 11syl2anc 585 . . . . 5 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝑐 β‰ˆ 𝐡)
1312ensymd 8948 . . . 4 (((Β¬ 𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) ∧ (𝑐 βŠ† 𝐴 ∧ 𝑐 β‰ˆ (cardβ€˜π΅))) β†’ 𝐡 β‰ˆ 𝑐)
14 bren 8896 . . . 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 6784 . . . . . 6 (𝑓:𝐡–1-1-onto→𝑐 β†’ 𝑓:𝐡–1-1→𝑐)
17 simplrl 776 . . . . . 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 6745 . . . . . 6 ((𝑓:𝐡–1-1→𝑐 ∧ 𝑐 βŠ† 𝐴) β†’ 𝑓:𝐡–1-1→𝐴)
1916, 17, 18syl2an2 685 . . . . 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 414 . . . 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 1921 . . 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 1939 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911   class class class wbr 5106  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  Ο‰com 7803   β‰ˆ cen 8883  Fincfn 8886  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880
This theorem is referenced by:  fidomtri  9934  hashdom  14285  erdsze2lem1  33854  eldioph2lem2  41127
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