Theorem subtsm 15267

Description: If A belongs to the smallest Tarski's class that contains X so does the subsets of A. CLASSES1. th. 6.

Assertion

Ref	Expression
subtsm	|- ((X e. B /\ A e. (tarskiMap` X)) -> ~PA C_ (tarskiMap` X))

Proof of Theorem subtsm

Step	Hyp	Ref	Expression
1		btmp 15252	. . . . . 6 |- ((X e. B /\ A e. (tarskiMap` X)) -> (A e. (tarskiMap` X) <-> A.t e. Tarski (X e. t -> A e. t)))
2		tarax1 15216	. . . . . . . . 9 |- ((t e. Tarski /\ A e. t) -> ~PA C_ t)
3	2	ex 402	. . . . . . . 8 |- (t e. Tarski -> (A e. t -> ~PA C_ t))
4	3	imim2d 28	. . . . . . 7 |- (t e. Tarski -> ((X e. t -> A e. t) -> (X e. t -> ~PA C_ t)))
5	4	ralimia 2166	. . . . . 6 |- (A.t e. Tarski (X e. t -> A e. t) -> A.t e. Tarski (X e. t -> ~PA C_ t))
6	1, 5	syl6bi 231	. . . . 5 |- ((X e. B /\ A e. (tarskiMap` X)) -> (A e. (tarskiMap` X) -> A.t e. Tarski (X e. t -> ~PA C_ t)))
7	6	ex 402	. . . 4 |- (X e. B -> (A e. (tarskiMap` X) -> (A e. (tarskiMap` X) -> A.t e. Tarski (X e. t -> ~PA C_ t))))
8	7	pm2.43d 79	. . 3 |- (X e. B -> (A e. (tarskiMap` X) -> A.t e. Tarski (X e. t -> ~PA C_ t)))
9	8	imp 377	. 2 |- ((X e. B /\ A e. (tarskiMap` X)) -> A.t e. Tarski (X e. t -> ~PA C_ t))
10		bpmp 15251	. . 3 |- ((X e. B /\ ~PA e. _V) -> (~PA C_ (tarskiMap` X) <-> A.t e. Tarski (X e. t -> ~PA C_ t)))
11		pwexg 3489	. . 3 |- (A e. (tarskiMap` X) -> ~PA e. _V)
12	10, 11	sylan2 500	. 2 |- ((X e. B /\ A e. (tarskiMap` X)) -> (~PA C_ (tarskiMap` X) <-> A.t e. Tarski (X e. t -> ~PA C_ t)))
13	9, 12	mpbird 213	1 |- ((X e. B /\ A e. (tarskiMap` X)) -> ~PA C_ (tarskiMap` X))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  ` cfv 3998   Tarski ctarski 15208  tarskiMapctarskim 15209

This theorem is referenced by:  vtarsuelt 15272

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-groth 10131

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-tsk 15210  df-tskmp 15248

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