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Mirrors > Home > NFE Home > Th. List > axprimlem1 | GIF version |
Description: Lemma for the primitive axioms. Primitive form of equality to a singleton. (Contributed by SF, 25-Mar-2015.) |
Ref | Expression |
---|---|
axprimlem1 | ⊢ (a = {B} ↔ ∀c(c ∈ a ↔ c = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2347 | . 2 ⊢ (a = {B} ↔ ∀c(c ∈ a ↔ c ∈ {B})) | |
2 | elsn 3748 | . . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 {csn 3737 |
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 3741 |
This theorem is referenced by: axprimlem2 4089 axsiprim 4093 axtyplowerprim 4094 axins2prim 4095 axins3prim 4096 snex 4111 |
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