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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 A:(α → ∗)
axext.2 B:(α → ∗)
Assertion
Ref Expression
axext ⊤⊧[(λx:α [(Ax:α) = (Bx:α)]) ⇒ [A = B]]
Distinct variable groups:   x,A   x,B   α,x

Proof of Theorem axext
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 axext.1 . . . . . 6 A:(α → ∗)
2 wv 58 . . . . . 6 x:α:α
31, 2wc 45 . . . . 5 (Ax:α):∗
43wl 59 . . . 4 λx:α (Ax:α):(α → ∗)
5 axext.2 . . . . . . . 8 B:(α → ∗)
65, 2wc 45 . . . . . . 7 (Bx:α):∗
73, 6weqi 68 . . . . . 6 [(Ax:α) = (Bx:α)]:∗
87ax4 140 . . . . 5 (λx:α [(Ax:α) = (Bx:α)])⊧[(Ax:α) = (Bx:α)]
9 wal 124 . . . . . 6 :((α → ∗) → ∗)
107wl 59 . . . . . 6 λx:α [(Ax:α) = (Bx:α)]:(α → ∗)
11 wv 58 . . . . . 6 y:α:α
129, 11ax-17 95 . . . . . 6 ⊤⊧[(λx:α y:α) = ]
137, 11ax-hbl1 93 . . . . . 6 ⊤⊧[(λx:α λx:α [(Ax:α) = (Bx:α)]y:α) = λx:α [(Ax:α) = (Bx:α)]]
149, 10, 11, 12, 13hbc 100 . . . . 5 ⊤⊧[(λx:α (λx:α [(Ax:α) = (Bx:α)])y:α) = (λx:α [(Ax:α) = (Bx:α)])]
153, 8, 14leqf 169 . . . 4 (λx:α [(Ax:α) = (Bx:α)])⊧[λx:α (Ax:α) = λx:α (Bx:α)]
1615ax-cb1 29 . . . . 5 (λx:α [(Ax:α) = (Bx:α)]):∗
171eta 166 . . . . 5 ⊤⊧[λx:α (Ax:α) = A]
1816, 17a1i 28 . . . 4 (λx:α [(Ax:α) = (Bx:α)])⊧[λx:α (Ax:α) = A]
195eta 166 . . . . 5 ⊤⊧[λx:α (Bx:α) = B]
2016, 19a1i 28 . . . 4 (λx:α [(Ax:α) = (Bx:α)])⊧[λx:α (Bx:α) = B]
214, 15, 18, 203eqtr3i 87 . . 3 (λx:α [(Ax:α) = (Bx:α)])⊧[A = B]
22 wtru 40 . . 3 ⊤:∗
2321, 22adantl 51 . 2 (⊤, (λx:α [(Ax:α) = (Bx:α)]))⊧[A = B]
2423ex 148 1 ⊤⊧[(λx:α [(Ax:α) = (Bx:α)]) ⇒ [A = B]]
Colors of variables: type var term
Syntax hints:  tv 1  ht 2  hb 3  kc 5  λkl 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
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