Theorem axprimlem1 4089

Description: Lemma for the primitive axioms. Primitive form of equality to a singleton. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axprimlem1	⊢ (a = {B} ↔ ∀c(c ∈ a ↔ c = B))

Distinct variable groups:   a,c   B,c

Allowed substitution hint:   B(a)

Proof of Theorem axprimlem1

Step	Hyp	Ref	Expression
1		dfcleq 2347	. 2 ⊢ (a = {B} ↔ ∀c(c ∈ a ↔ c ∈ {B}))
2		elsn 3749	. . . 4 ⊢ (c ∈ {B} ↔ c = B)
3	2	bibi2i 304	. . 3 ⊢ ((c ∈ a ↔ c ∈ {B}) ↔ (c ∈ a ↔ c = B))
4	3	albii 1566	. 2 ⊢ (∀c(c ∈ a ↔ c ∈ {B}) ↔ ∀c(c ∈ a ↔ c = B))
5	1, 4	bitri 240	1 ⊢ (a = {B} ↔ ∀c(c ∈ a ↔ c = B))

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sn 3742

This theorem is referenced by:  axprimlem2  4090  axsiprim  4094  axtyplowerprim  4095  axins2prim  4096  axins3prim  4097  snex  4112