Theorem inf3lem3 4624

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 4629 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 4605.

Hypotheses

Ref	Expression
inf3lem.1	|- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
inf3lem.2	|- F = (rec(G, (/)) |` om)
inf3lem.3	|- A e. V
inf3lem.4	|- B e. V

Assertion

Ref	Expression
inf3lem3	|- ((x =/= (/) /\ x (_ U.x) -> (A e. om -> (F` A) =/= (F` suc A)))

Distinct variable group:   x,y,z,w

Proof of Theorem inf3lem3

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . . 7 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
2		inf3lem.2	. . . . . . 7 |- F = (rec(G, (/)) |` om)
3		inf3lem.3	. . . . . . 7 |- A e. V
4		inf3lem.4	. . . . . . 7 |- B e. V
5	1, 2, 3, 4	inf3lem2 4623	. . . . . 6 |- ((x =/= (/) /\ x (_ U.x) -> (A e. om -> (F` A) =/= x))
6	5	com12 11	. . . . 5 |- (A e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` A) =/= x))
7	1, 2, 3, 4	inf3lemd 4621	. . . . 5 |- (A e. om -> (F` A) (_ x)
8	6, 7	jctild 603	. . . 4 |- (A e. om -> ((x =/= (/) /\ x (_ U.x) -> ((F` A) (_ x /\ (F` A) =/= x)))
9		pssdifn0 2333	. . . 4 |- (((F` A) (_ x /\ (F` A) =/= x) -> (x \ (F` A)) =/= (/))
10	8, 9	syl6 22	. . 3 |- (A e. om -> ((x =/= (/) /\ x (_ U.x) -> (x \ (F` A)) =/= (/)))
11	1, 2, 3, 4	inf3lemc 4620	. . . . . . . . . 10 |- (A e. om -> (F` suc A) = (G` (F` A)))
12	11	eleq2d 1544	. . . . . . . . 9 |- (A e. om -> (v e. (F` suc A) <-> v e. (G` (F` A))))
13		eldifi 2165	. . . . . . . . . . 11 |- (v e. (x \ (F` A)) -> v e. x)
14		inssdif0 2337	. . . . . . . . . . . 12 |- ((v i^i x) (_ (F` A) <-> (v i^i (x \ (F` A))) = (/))
15	14	biimpr 152	. . . . . . . . . . 11 |- ((v i^i (x \ (F` A))) = (/) -> (v i^i x) (_ (F` A))
16	13, 15	anim12i 333	. . . . . . . . . 10 |- ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> (v e. x /\ (v i^i x) (_ (F` A)))
17		visset 1816	. . . . . . . . . . 11 |- v e. V
18		fvex 3738	. . . . . . . . . . 11 |- (F` A) e. V
19	1, 2, 17, 18	inf3lema 4618	. . . . . . . . . 10 |- (v e. (G` (F` A)) <-> (v e. x /\ (v i^i x) (_ (F` A)))
20	16, 19	sylibr 200	. . . . . . . . 9 |- ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> v e. (G` (F` A)))
21	12, 20	syl5bir 210	. . . . . . . 8 |- (A e. om -> ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> v e. (F` suc A)))
22		eldifn 2166	. . . . . . . . . 10 |- (v e. (x \ (F` A)) -> -. v e. (F` A))
23	22	adantr 391	. . . . . . . . 9 |- ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> -. v e. (F` A))
24	23	a1i 8	. . . . . . . 8 |- (A e. om -> ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> -. v e. (F` A)))
25	21, 24	jcad 602	. . . . . . 7 |- (A e. om -> ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> (v e. (F` suc A) /\ -. v e. (F` A))))
26		eleq2 1538	. . . . . . . . . 10 |- ((F` A) = (F` suc A) -> (v e. (F` A) <-> v e. (F` suc A)))
27	26	biimprd 154	. . . . . . . . 9 |- ((F` A) = (F` suc A) -> (v e. (F` suc A) -> v e. (F` A)))
28		iman 237	. . . . . . . . 9 |- ((v e. (F` suc A) -> v e. (F` A)) <-> -. (v e. (F` suc A) /\ -. v e. (F` A)))
29	27, 28	sylib 198	. . . . . . . 8 |- ((F` A) = (F` suc A) -> -. (v e. (F` suc A) /\ -. v e. (F` A)))
30	29	necon2ai 1614	. . . . . . 7 |- ((v e. (F` suc A) /\ -. v e. (F` A)) -> (F` A) =/= (F` suc A))
31	25, 30	syl6 22	. . . . . 6 |- (A e. om -> ((v e. (x \ (F` A)) /\ (v i^i (x \ (F` A))) = (/)) -> (F` A) =/= (F` suc A)))
32	31	exp3a 376	. . . . 5 |- (A e. om -> (v e. (x \ (F` A)) -> ((v i^i (x \ (F` A))) = (/) -> (F` A) =/= (F` suc A))))
33	32	r19.23adv 1749	. . . 4 |- (A e. om -> (E.v e. (x \ (F` A))(v i^i (x \ (F` A))) = (/) -> (F` A) =/= (F` suc A)))
34		visset 1816	. . . . . 6 |- x e. V
35		difss 2170	. . . . . 6 |- (x \ (F` A)) (_ x
36	34, 35	ssexi 2725	. . . . 5 |- (x \ (F` A)) e. V
37	36	zfreg 4605	. . . 4 |- ((x \ (F` A)) =/= (/) -> E.v e. (x \ (F` A))(v i^i (x \ (F` A))) = (/))
38	33, 37	syl5 21	. . 3 |- (A e. om -> ((x \ (F` A)) =/= (/) -> (F` A) =/= (F` suc A)))
39	10, 38	syld 27	. 2 |- (A e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` A) =/= (F` suc A)))
40	39	com12 11	1 |- ((x =/= (/) /\ x (_ U.x) -> (A e. om -> (F` A) =/= (F` suc A)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  E.wrex 1649  {crab 1651  Vcvv 1814   \ cdif 2047   i^i cin 2049   (_ wss 2050  (/)c0 2283  U.cuni 2507  {copab 2671  suc csuc 2956  omcom 3137   |` cres 3178  ` cfv 3188  reccrdg 3937

This theorem is referenced by:  inf3lem4 4625

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938

Copyright terms: Public domain