Theorem inf3lem5 4626

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 4629 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:   x,y,z,w

Proof of Theorem inf3lem5

Step	Hyp	Ref	Expression
1		elnn 3148	. . . 4 |- ((B e. A /\ A e. om) -> B e. om)
2	1	ancoms 438	. . 3 |- ((A e. om /\ B e. A) -> B e. om)
3		nnord 3146	. . . . . . . . 9 |- (A e. om -> Ord A)
4		ordsucss 3075	. . . . . . . . 9 |- (Ord A -> (B e. A -> suc B (_ A))
5	3, 4	syl 10	. . . . . . . 8 |- (A e. om -> (B e. A -> suc B (_ A))
6	5	adantr 391	. . . . . . 7 |- ((A e. om /\ B e. om) -> (B e. A -> suc B (_ A))
7		fveq2 3730	. . . . . . . . . . . 12 |- (v = suc B -> (F` v) = (F` suc B))
8	7	psseq2d 2144	. . . . . . . . . . 11 |- (v = suc B -> ((F` B) (. (F` v) <-> (F` B) (. (F` suc B)))
9	8	imbi2d 614	. . . . . . . . . 10 |- (v = suc B -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc B))))
10		fveq2 3730	. . . . . . . . . . . 12 |- (v = u -> (F` v) = (F` u))
11	10	psseq2d 2144	. . . . . . . . . . 11 |- (v = u -> ((F` B) (. (F` v) <-> (F` B) (. (F` u)))
12	11	imbi2d 614	. . . . . . . . . 10 |- (v = u -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` u))))
13		fveq2 3730	. . . . . . . . . . . 12 |- (v = suc u -> (F` v) = (F` suc u))
14	13	psseq2d 2144	. . . . . . . . . . 11 |- (v = suc u -> ((F` B) (. (F` v) <-> (F` B) (. (F` suc u)))
15	14	imbi2d 614	. . . . . . . . . 10 |- (v = suc u -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
16		fveq2 3730	. . . . . . . . . . . 12 |- (v = A -> (F` v) = (F` A))
17	16	psseq2d 2144	. . . . . . . . . . 11 |- (v = A -> ((F` B) (. (F` v) <-> (F` B) (. (F` A)))
18	17	imbi2d 614	. . . . . . . . . 10 |- (v = A -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
19		peano2b 3153	. . . . . . . . . . 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 4625	. . . . . . . . . . . 12 |- ((x =/= (/) /\ x (_ U.x) -> (B e. om -> (F` B) (. (F` suc B)))
24	23	com12 11	. . . . . . . . . . 11 |- (B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc B)))
25	19, 24	sylbir 201	. . . . . . . . . 10 |- (suc B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc B)))
26		visset 1816	. . . . . . . . . . . . . 14 |- u e. V
27	20, 21, 26, 22	inf3lem4 4625	. . . . . . . . . . . . 13 |- ((x =/= (/) /\ x (_ U.x) -> (u e. om -> (F` u) (. (F` suc u)))
28		psstr 2153	. . . . . . . . . . . . . 14 |- (((F` B) (. (F` u) /\ (F` u) (. (F` suc u)) -> (F` B) (. (F` suc u))
29	28	expcom 374	. . . . . . . . . . . . 13 |- ((F` u) (. (F` suc u) -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u)))
30	27, 29	syl6com 53	. . . . . . . . . . . 12 |- (u e. om -> ((x =/= (/) /\ x (_ U.x) -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u))))
31	30	a2d 13	. . . . . . . . . . 11 |- (u e. om -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` u)) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
32	31	ad2antrr 406	. . . . . . . . . 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))))
33	9, 12, 15, 18, 25, 32	findsg 3163	. . . . . . . . 9 |- (((A e. om /\ suc B e. om) /\ suc B (_ A) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))
34	33	ex 373	. . . . . . . 8 |- ((A e. om /\ suc B e. om) -> (suc B (_ A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
35	34, 19	sylan2b 454	. . . . . . 7 |- ((A e. om /\ B e. om) -> (suc B (_ A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
36	6, 35	syld 27	. . . . . 6 |- ((A e. om /\ B e. om) -> (B e. A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
37	36	ex 373	. . . . 5 |- (A e. om -> (B e. om -> (B e. A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))))
38	37	com23 32	. . . 4 |- (A e. om -> (B e. A -> (B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))))
39	38	imp 350	. . 3 |- ((A e. om /\ B e. A) -> (B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
40	2, 39	mpd 26	. 2 |- ((A e. om /\ B e. A) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))
41	40	com12 11	1 |- ((x =/= (/) /\ x (_ U.x) -> ((A e. om /\ B e. A) -> (F` B) (. (F` A)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  {crab 1651  Vcvv 1814   i^i cin 2049   (_ wss 2050   (. wpss 2051  (/)c0 2283  U.cuni 2507  {copab 2671  Ord word 2953  suc csuc 2956  omcom 3137   |` cres 3178  ` cfv 3188  reccrdg 3937

This theorem is referenced by:  inf3lem6 4627

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

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