Theorem inf3lem5 3468

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 3471 for detailed description.

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
inf3lem5	|- ((-. x = (/) /\ x (_ U.x) -> ((A e. om /\ B e. A) -> (F` B) (. (F` A)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lem5

Step	Hyp	Ref	Expression
1		elnn 2383	. . . 4 |- ((B e. A /\ A e. om) -> B e. om)
2	1	ancoms 334	. . 3 |- ((A e. om /\ B e. A) -> B e. om)
3		nnord 2381	. . . . . . . . 9 |- (A e. om -> Ord A)
4		ordsucss 2320	. . . . . . . . 9 |- (Ord A -> (B e. A -> suc B (_ A))
5	3, 4	syl 12	. . . . . . . 8 |- (A e. om -> (B e. A -> suc B (_ A))
6	5	adantr 306	. . . . . . 7 |- ((A e. om /\ B e. om) -> (B e. A -> suc B (_ A))
7		fveq2 2832	. . . . . . . . . . . 12 |- (v = suc B -> (F` v) = (F` suc B))
8	7	psseq2d 1565	. . . . . . . . . . 11 |- (v = suc B -> ((F` B) (. (F` v) <-> (F` B) (. (F` suc B)))
9	8	imbi2d 464	. . . . . . . . . 10 |- (v = suc B -> (((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` suc B))))
10		fveq2 2832	. . . . . . . . . . . 12 |- (v = u -> (F` v) = (F` u))
11	10	psseq2d 1565	. . . . . . . . . . 11 |- (v = u -> ((F` B) (. (F` v) <-> (F` B) (. (F` u)))
12	11	imbi2d 464	. . . . . . . . . 10 |- (v = u -> (((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` u))))
13		fveq2 2832	. . . . . . . . . . . 12 |- (v = suc u -> (F` v) = (F` suc u))
14	13	psseq2d 1565	. . . . . . . . . . 11 |- (v = suc u -> ((F` B) (. (F` v) <-> (F` B) (. (F` suc u)))
15	14	imbi2d 464	. . . . . . . . . 10 |- (v = suc u -> (((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
16		fveq2 2832	. . . . . . . . . . . 12 |- (v = A -> (F` v) = (F` A))
17	16	psseq2d 1565	. . . . . . . . . . 11 |- (v = A -> ((F` B) (. (F` v) <-> (F` B) (. (F` A)))
18	17	imbi2d 464	. . . . . . . . . 10 |- (v = A -> (((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
19		peano2b 2388	. . . . . . . . . . 11 |- (B e. om <-> suc B e. om)
20		inf3lem.1	. . . . . . . . . . . . 13 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
21		inf3lem.2	. . . . . . . . . . . . 13 |- F = (rec(G, (/)) |` om)
22		inf3lem.4	. . . . . . . . . . . . 13 |- B e. V
23	20, 21, 22, 22	inf3lem4 3467	. . . . . . . . . . . 12 |- ((-. x = (/) /\ x (_ U.x) -> (B e. om -> (F` B) (. (F` suc B)))
24	23	com12 13	. . . . . . . . . . 11 |- (B e. om -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` suc B)))
25	19, 24	sylbir 176	. . . . . . . . . 10 |- (suc B e. om -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` suc B)))
26		visset 1350	. . . . . . . . . . . . . . 15 |- u e. V
27	20, 21, 26, 22	inf3lem4 3467	. . . . . . . . . . . . . 14 |- ((-. x = (/) /\ x (_ U.x) -> (u e. om -> (F` u) (. (F` suc u)))
28		psstr 1574	. . . . . . . . . . . . . . . 16 |- (((F` B) (. (F` u) /\ (F` u) (. (F` suc u)) -> (F` B) (. (F` suc u))
29	28	exp 291	. . . . . . . . . . . . . . 15 |- ((F` B) (. (F` u) -> ((F` u) (. (F` suc u) -> (F` B) (. (F` suc u)))
30	29	com12 13	. . . . . . . . . . . . . 14 |- ((F` u) (. (F` suc u) -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u)))
31	27, 30	syl6 23	. . . . . . . . . . . . 13 |- ((-. x = (/) /\ x (_ U.x) -> (u e. om -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u))))
32	31	com12 13	. . . . . . . . . . . 12 |- (u e. om -> ((-. x = (/) /\ x (_ U.x) -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u))))
33	32	a2d 15	. . . . . . . . . . 11 |- (u e. om -> (((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` u)) -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
34	33	ad2antll 320	. . . . . . . . . 10 |- (((u e. om /\ suc B e. om) /\ suc B (_ u) -> (((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` u)) -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
35	9, 12, 15, 18, 25, 34	findsg 2398	. . . . . . . . 9 |- (((A e. om /\ suc B e. om) /\ suc B (_ A) -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A)))
36	35	exp 291	. . . . . . . 8 |- ((A e. om /\ suc B e. om) -> (suc B (_ A -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
37	36, 19	sylan2b 347	. . . . . . 7 |- ((A e. om /\ B e. om) -> (suc B (_ A -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
38	6, 37	syld 27	. . . . . 6 |- ((A e. om /\ B e. om) -> (B e. A -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
39	38	exp 291	. . . . 5 |- (A e. om -> (B e. om -> (B e. A -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A)))))
40	39	com23 32	. . . 4 |- (A e. om -> (B e. A -> (B e. om -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A)))))
41	40	imp 277	. . 3 |- ((A e. om /\ B e. A) -> (B e. om -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
42	2, 41	mpd 46	. 2 |- ((A e. om /\ B e. A) -> ((-. x = (/) /\ x (_ U.x) -> (F` B) (. (F` A)))
43	42	com12 13	1 |- ((-. x = (/) /\ x (_ U.x) -> ((A e. om /\ B e. A) -> (F` B) (. (F` A)))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348   i^i cin 1486   (_ wss 1487   (. wpss 1488  (/)c0 1707  U.cuni 1919  {copab 2055  Ord word 2198  suc csuc 2201  omcom 2372   |` cres 2412  ` cfv 2422  reccrdg 2969

This theorem is referenced by:  inf3lem6 3469

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970

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