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Theorem axext 206

Description: Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461.

Hypotheses

Ref	Expression
axext.1
axext.2

Assertion

Ref	Expression
axext

Distinct variable groups:   ,   ,   ,

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Proof of Theorem axext Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axext.1	. . . . . 6
2		wv 58	. . . . . 6
3	1, 2	wc 45	. . . . 5
4	3	wl 59	. . . 4
5		axext.2	. . . . . . . 8
6	5, 2	wc 45	. . . . . . 7
7	3, 6	weqi 68	. . . . . 6
8	7	ax4 140	. . . . 5
9		wal 124	. . . . . 6
10	7	wl 59	. . . . . 6
11		wv 58	. . . . . 6
12	9, 11	ax-17 95	. . . . . 6
13	7, 11	ax-hbl1 93	. . . . . 6
14	9, 10, 11, 12, 13	hbc 100	. . . . 5
15	3, 8, 14	leqf 169	. . . 4
16	15	ax-cb1 29	. . . . 5
17	1	eta 166	. . . . 5
18	16, 17	a1i 28	. . . 4
19	5	eta 166	. . . . 5
20	16, 19	a1i 28	. . . 4
21	4, 15, 18, 20	3eqtr3i 87	. . 3
22		wtru 40	. . 3
23	21, 22	adantl 51	. 2
24	23	ex 148	1

Colors of variables: type var term

Syntax hints:  tv 1   ht 2  hb 3  kc 5  kl 6   ke 7  kt 8  kbr 9   wffMMJ2 11  wffMMJ2t 12   tim 111  tal 112

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119

This theorem is referenced by: (None)

Copyright terms: Public domain

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