New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > tfin0c | GIF version |
Description: The finite T operator is fixed at 0c. (Contributed by SF, 29-Jan-2015.) |
Ref | Expression |
---|---|
tfin0c | ⊢ Tfin 0c = 0c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 4402 | . . 3 ⊢ 0c ∈ Nn | |
2 | tfincl 4492 | . . 3 ⊢ (0c ∈ Nn → Tfin 0c ∈ Nn ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Tfin 0c ∈ Nn |
4 | pw10 4161 | . . 3 ⊢ ℘1∅ = ∅ | |
5 | nulel0c 4422 | . . . 4 ⊢ ∅ ∈ 0c | |
6 | tfinpw1 4494 | . . . 4 ⊢ ((0c ∈ Nn ∧ ∅ ∈ 0c) → ℘1∅ ∈ Tfin 0c) | |
7 | 1, 5, 6 | mp2an 653 | . . 3 ⊢ ℘1∅ ∈ Tfin 0c |
8 | 4, 7 | eqeltrri 2424 | . 2 ⊢ ∅ ∈ Tfin 0c |
9 | nnceleq 4430 | . 2 ⊢ ((( Tfin 0c ∈ Nn ∧ 0c ∈ Nn ) ∧ (∅ ∈ Tfin 0c ∧ ∅ ∈ 0c)) → Tfin 0c = 0c) | |
10 | 3, 1, 8, 5, 9 | mp4an 654 | 1 ⊢ Tfin 0c = 0c |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ∅c0 3550 ℘1cpw1 4135 Nn cnnc 4373 0cc0c 4374 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 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-tfin 4443 |
This theorem is referenced by: tfin1c 4499 sfintfin 4532 tfinnn 4534 |
Copyright terms: Public domain | W3C validator |