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Mirrors > Home > NFE Home > Th. List > tfinex | GIF version |
Description: The finite T operator is always a set. (Contributed by SF, 26-Jan-2015.) |
Ref | Expression |
---|---|
tfinex | ⊢ Tfin A ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tfin 4443 | . 2 ⊢ Tfin A = if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) | |
2 | 0ex 4110 | . . 3 ⊢ ∅ ∈ V | |
3 | iotaex 4356 | . . 3 ⊢ (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)) ∈ V | |
4 | 2, 3 | ifex 3720 | . 2 ⊢ if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) ∈ V |
5 | 1, 4 | eqeltri 2423 | 1 ⊢ Tfin A ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 Vcvv 2859 ∅c0 3550 ifcif 3662 ℘1cpw1 4135 ℩cio 4337 Nn cnnc 4373 Tfin ctfin 4435 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 ax-nin 4078 |
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-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-if 3663 df-sn 3741 df-pr 3742 df-uni 3892 df-iota 4339 df-tfin 4443 |
This theorem is referenced by: tfinltfinlem1 4500 tfinltfin 4501 eventfin 4517 oddtfin 4518 sfintfinlem1 4531 tfinnnlem1 4533 tfinnn 4534 vfinspsslem1 4550 |
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