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Mirrors > Home > NFE Home > Th. List > int0 | GIF version |
Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
int0 | ⊢ ∩∅ = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3554 | . . . . . 6 ⊢ ¬ y ∈ ∅ | |
2 | 1 | pm2.21i 123 | . . . . 5 ⊢ (y ∈ ∅ → x ∈ y) |
3 | 2 | ax-gen 1546 | . . . 4 ⊢ ∀y(y ∈ ∅ → x ∈ y) |
4 | eqid 2353 | . . . 4 ⊢ x = x | |
5 | 3, 4 | 2th 230 | . . 3 ⊢ (∀y(y ∈ ∅ → x ∈ y) ↔ x = x) |
6 | 5 | abbii 2465 | . 2 ⊢ {x ∣ ∀y(y ∈ ∅ → x ∈ y)} = {x ∣ x = x} |
7 | df-int 3927 | . 2 ⊢ ∩∅ = {x ∣ ∀y(y ∈ ∅ → x ∈ y)} | |
8 | df-v 2861 | . 2 ⊢ V = {x ∣ x = x} | |
9 | 6, 7, 8 | 3eqtr4i 2383 | 1 ⊢ ∩∅ = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 = wceq 1642 ∈ wcel 1710 {cab 2339 Vcvv 2859 ∅c0 3550 ∩cint 3926 |
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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 df-int 3927 |
This theorem is referenced by: unissint 3950 uniintsn 3963 rint0 3966 |
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