Theorem dftr2 2043

Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40.

Assertion

Ref	Expression
dftr2	⊢ (Tr A ↔ ∀x∀y((x ∈ y ∧ y ∈ A) → x ∈ A))

Distinct variable group(s):   x,y,A

Proof of Theorem dftr2

Step	Hyp	Ref	Expression
1	â	dfss2 1497	. 2 ⊢ (∪A ⊆ A ↔ ∀x(x ∈ ∪A → x ∈ A))
2		df-tr 2042	. 2 ⊢ (Tr A ↔ ∪A ⊆ A)
3		19.23v 950	. . . 4 ⊢ (∀y((x ∈ y ∧ y ∈ A) → x ∈ A) ↔ (∃y(x ∈ y ∧ y ∈ A) → x ∈ A))
4		eluni 1922	. . . . 5 ⊢ (x ∈ ∪A ↔ ∃y(x ∈ y ∧ y ∈ A))
5	4	imbi1i 161	. . . 4 ⊢ ((x ∈ ∪A → x ∈ A) ↔ (∃y(x ∈ y ∧ y ∈ A) → x ∈ A))
6	3, 5	bitr4 154	. . 3 ⊢ (∀y((x ∈ y ∧ y ∈ A) → x ∈ A) ↔ (x ∈ ∪A → x ∈ A))
7	6	bial 695	. 2 ⊢ (∀x∀y((x ∈ y ∧ y ∈ A) → x ∈ A) ↔ ∀x(x ∈ ∪A → x ∈ A))
8	1, 2, 7	3bitr4 158	1 ⊢ (Tr A ↔ ∀x∀y((x ∈ y ∧ y ∈ A) → x ∈ A))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   ∈ wcel 1092   ⊆ wss 1487  ∪cuni 1919  Tr wtr 2041

This theorem is referenced by:  dftr5 2044  trel 2048  ordelord 2221  ordom 2382  tfrlem8 2956  trcl 3489  ondomon 3662

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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