Theorem tz9.13 4643

Description: Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78.

Hypothesis

Ref	Expression
tz9.13.1	|- A e. V

Assertion

Ref	Expression
tz9.13	|- E.x e. On A e. (R1` x)

Distinct variable group:   x,A

Proof of Theorem tz9.13

Step	Hyp	Ref	Expression
1		tz9.13.1	. . 3 |- A e. V
2		setind 4628	. . . 4 |- (A.z(z (_ {y | E.x e. On y e. (R1` x)} -> z e. {y | E.x e. On y e. (R1` x)}) -> {y | E.x e. On y e. (R1` x)} = V)
3		ssel 2059	. . . . . . . 8 |- (z (_ {y | E.x e. On y e. (R1` x)} -> (w e. z -> w e. {y | E.x e. On y e. (R1` x)}))
4		visset 1809	. . . . . . . . 9 |- w e. V
5		eleq1 1531	. . . . . . . . . 10 |- (y = w -> (y e. (R1` x) <-> w e. (R1` x)))
6	5	rexbidv 1661	. . . . . . . . 9 |- (y = w -> (E.x e. On y e. (R1` x) <-> E.x e. On w e. (R1` x)))
7	4, 6	elab 1893	. . . . . . . 8 |- (w e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On w e. (R1` x))
8	3, 7	syl6ib 212	. . . . . . 7 |- (z (_ {y | E.x e. On y e. (R1` x)} -> (w e. z -> E.x e. On w e. (R1` x)))
9	8	r19.21aiv 1710	. . . . . 6 |- (z (_ {y | E.x e. On y e. (R1` x)} -> A.w e. z E.x e. On w e. (R1` x))
10		visset 1809	. . . . . . 7 |- z e. V
11	10	tz9.12 4642	. . . . . 6 |- (A.w e. z E.x e. On w e. (R1` x) -> E.x e. On z e. (R1` x))
12	9, 11	syl 10	. . . . 5 |- (z (_ {y | E.x e. On y e. (R1` x)} -> E.x e. On z e. (R1` x))
13		eleq1 1531	. . . . . . 7 |- (y = z -> (y e. (R1` x) <-> z e. (R1` x)))
14	13	rexbidv 1661	. . . . . 6 |- (y = z -> (E.x e. On y e. (R1` x) <-> E.x e. On z e. (R1` x)))
15	10, 14	elab 1893	. . . . 5 |- (z e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On z e. (R1` x))
16	12, 15	sylibr 200	. . . 4 |- (z (_ {y | E.x e. On y e. (R1` x)} -> z e. {y | E.x e. On y e. (R1` x)})
17	2, 16	mpg 984	. . 3 |- {y | E.x e. On y e. (R1` x)} = V
18	1, 17	eleqtrr 1544	. 2 |- A e. {y | E.x e. On y e. (R1` x)}
19		eleq1 1531	. . . 4 |- (y = A -> (y e. (R1` x) <-> A e. (R1` x)))
20	19	rexbidv 1661	. . 3 |- (y = A -> (E.x e. On y e. (R1` x) <-> E.x e. On A e. (R1` x)))
21	1, 20	elab 1893	. 2 |- (A e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On A e. (R1` x))
22	18, 21	mpbi 189	1 |- E.x e. On A e. (R1` x)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  {cab 1461  A.wral 1642  E.wrex 1643  Vcvv 1807   (_ wss 2043  Oncon0 2943  ` cfv 3177  R1cr1 4621

This theorem is referenced by:  tz9.13g 4644  jech9.3 4646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623

Copyright terms: Public domain