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Mirrors > Home > NFE Home > Th. List > tfinnul | GIF version |
Description: The finite T operator applied to the empty set is empty. Theorem X.1.29 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
tfinnul | ⊢ Tfin ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tfin 4443 | . 2 ⊢ Tfin ∅ = if(∅ = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ ∅ ℘1y ∈ x))) | |
2 | eqid 2353 | . . 3 ⊢ ∅ = ∅ | |
3 | iftrue 3668 | . . 3 ⊢ (∅ = ∅ → if(∅ = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ ∅ ℘1y ∈ x))) = ∅) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ if(∅ = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ ∅ ℘1y ∈ x))) = ∅ |
5 | 1, 4 | eqtri 2373 | 1 ⊢ Tfin ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∅c0 3550 ifcif 3662 ℘1cpw1 4135 ℩cio 4337 Nn cnnc 4373 Tfin ctfin 4435 |
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-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-if 3663 df-tfin 4443 |
This theorem is referenced by: tfincl 4492 tfin11 4493 tfinltfinlem1 4500 tfinltfin 4501 vfinncvntnn 4548 |
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