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Theorem isfiniteg 7317
 Description: A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
isfiniteg

Proof of Theorem isfiniteg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isfi 7081 . . 3
2 nnsdomg 7316 . . . . 5
3 sdomen1 7201 . . . . 5
42, 3syl5ibrcom 214 . . . 4
54rexlimdva 2787 . . 3
61, 5syl5bi 209 . 2
7 isfinite2 7315 . 2
86, 7impbid1 195 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wcel 1721  wrex 2664  cvv 2913   class class class wbr 4167  com 4799   cen 7056   csdm 7058  cfn 7059 This theorem is referenced by:  unfi2  7326  unifi2  7346  isfinite  7554  axcclem  8284 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382  ax-sep 4285  ax-nul 4293  ax-pow 4332  ax-pr 4358  ax-un 4655 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2256  df-mo 2257  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-ne 2566  df-ral 2668  df-rex 2669  df-reu 2670  df-rab 2672  df-v 2915  df-sbc 3119  df-csb 3209  df-dif 3280  df-un 3282  df-in 3284  df-ss 3291  df-pss 3293  df-nul 3586  df-if 3697  df-pw 3758  df-sn 3777  df-pr 3778  df-tp 3779  df-op 3780  df-uni 3972  df-int 4007  df-iun 4051  df-br 4168  df-opab 4222  df-mpt 4223  df-tr 4258  df-eprel 4449  df-id 4453  df-po 4458  df-so 4459  df-fr 4496  df-we 4498  df-ord 4539  df-on 4540  df-lim 4541  df-suc 4542  df-om 4800  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5372  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-recs 6583  df-rdg 6618  df-er 6855  df-en 7060  df-dom 7061  df-sdom 7062  df-fin 7063
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