New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > tfinlefin | GIF version |
Description: Ordering rule for the finite T operation. Theorem X.1.33 of [Rosser] p. 529. (Contributed by SF, 2-Feb-2015.) |
Ref | Expression |
---|---|
tfinlefin | ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ ≤fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ ≤fin )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfinltfin 4501 | . . . 4 ⊢ ((N ∈ Nn ∧ M ∈ Nn ) → (⟪N, M⟫ ∈ <fin ↔ ⟪ Tfin N, Tfin M⟫ ∈ <fin )) | |
2 | 1 | ancoms 439 | . . 3 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪N, M⟫ ∈ <fin ↔ ⟪ Tfin N, Tfin M⟫ ∈ <fin )) |
3 | 2 | notbid 285 | . 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (¬ ⟪N, M⟫ ∈ <fin ↔ ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin )) |
4 | lenltfin 4469 | . 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ ≤fin ↔ ¬ ⟪N, M⟫ ∈ <fin )) | |
5 | tfincl 4492 | . . 3 ⊢ (M ∈ Nn → Tfin M ∈ Nn ) | |
6 | tfincl 4492 | . . 3 ⊢ (N ∈ Nn → Tfin N ∈ Nn ) | |
7 | lenltfin 4469 | . . 3 ⊢ (( Tfin M ∈ Nn ∧ Tfin N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ ≤fin ↔ ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin )) | |
8 | 5, 6, 7 | syl2an 463 | . 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪ Tfin M, Tfin N⟫ ∈ ≤fin ↔ ¬ ⟪ Tfin N, Tfin M⟫ ∈ <fin )) |
9 | 3, 4, 8 | 3bitr4d 276 | 1 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (⟪M, N⟫ ∈ ≤fin ↔ ⟪ Tfin M, Tfin N⟫ ∈ ≤fin )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ⟪copk 4057 Nn cnnc 4373 ≤fin clefin 4432 <fin cltfin 4433 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 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-3or 935 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-lefin 4440 df-ltfin 4441 df-tfin 4443 |
This theorem is referenced by: vfintle 4546 |
Copyright terms: Public domain | W3C validator |