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Theorem axgroth6 8704
 Description: The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set , there exists a set containing , the subsets of the members of , the power sets of the members of , and the subsets of of cardinality less than that of . (Contributed by NM, 21-Jun-2009.)
Assertion
Ref Expression
axgroth6
Distinct variable group:   ,,

Proof of Theorem axgroth6
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 8700 . 2
2 biid 229 . . . 4
3 pweq 3803 . . . . . . . . 9
43sseq1d 3376 . . . . . . . 8
54cbvralv 2933 . . . . . . 7
6 ssid 3368 . . . . . . . . . 10
7 sseq2 3371 . . . . . . . . . . 11
87rspcev 3053 . . . . . . . . . 10
96, 8mpan2 654 . . . . . . . . 9
10 pweq 3803 . . . . . . . . . . . . 13
1110sseq1d 3376 . . . . . . . . . . . 12
1211rspccv 3050 . . . . . . . . . . 11
13 pwss 3814 . . . . . . . . . . . 12
14 vex 2960 . . . . . . . . . . . . . 14
1514pwex 4383 . . . . . . . . . . . . 13
16 sseq1 3370 . . . . . . . . . . . . . 14
17 eleq1 2497 . . . . . . . . . . . . . 14
1816, 17imbi12d 313 . . . . . . . . . . . . 13
1915, 18spcv 3043 . . . . . . . . . . . 12
2013, 19sylbi 189 . . . . . . . . . . 11
2112, 20syl6 32 . . . . . . . . . 10
2221rexlimdv 2830 . . . . . . . . 9
239, 22impbid2 197 . . . . . . . 8
2423ralbidv 2726 . . . . . . 7
255, 24sylbi 189 . . . . . 6
2625pm5.32i 620 . . . . 5
27 r19.26 2839 . . . . 5
28 r19.26 2839 . . . . 5
2926, 27, 283bitr4i 270 . . . 4
3014elpw 3806 . . . . . 6
31 impexp 435 . . . . . . . . 9
32 vex 2960 . . . . . . . . . . . 12
33 ssdomg 7154 . . . . . . . . . . . 12
3432, 33ax-mp 8 . . . . . . . . . . 11
3534pm4.71i 615 . . . . . . . . . 10
3635imbi1i 317 . . . . . . . . 9
37 brsdom 7131 . . . . . . . . . . . 12
3837imbi1i 317 . . . . . . . . . . 11
39 impexp 435 . . . . . . . . . . 11
4038, 39bitri 242 . . . . . . . . . 10
4140imbi2i 305 . . . . . . . . 9
4231, 36, 413bitr4ri 271 . . . . . . . 8
4342pm5.74ri 239 . . . . . . 7
44 pm4.64 363 . . . . . . 7
4543, 44syl6bb 254 . . . . . 6
4630, 45sylbi 189 . . . . 5
4746ralbiia 2738 . . . 4
482, 29, 473anbi123i 1143 . . 3
4948exbii 1593 . 2
501, 49mpbir 202 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wo 359   wa 360   w3a 937  wal 1550  wex 1551   wceq 1653   wcel 1726  wral 2706  wrex 2707  cvv 2957   wss 3321  cpw 3800   class class class wbr 4213   cen 7107   cdom 7108   csdm 7109 This theorem is referenced by:  grothomex  8705  grothac  8706 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-groth 8699 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-dom 7112  df-sdom 7113
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