Theorem dftr3 4756

Description: An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)

Assertion

Ref	Expression
dftr3	⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem dftr3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr5 4755	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
2		dfss3 3592	. . 3 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
3	2	ralbii 2980	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
4	1, 3	bitr4i 267	1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)

Syntax hints:   ↔ wb 196   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753

This theorem is referenced by:  trss  4761  trssOLD  4762  trin  4763  triun  4766  trint  4768  tron  5746  ssorduni  6985  suceloni  7013  dfrecs3  7469  ordtypelem2  8424  tcwf  8746  itunitc  9243  wunex2  9560  wfgru  9638