New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > opkltfing | GIF version |
Description: Kuratowski ordered pair membership in finite less than. (Contributed by SF, 27-Jan-2015.) |
Ref | Expression |
---|---|
opkltfing | ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ <fin ↔ (A ≠ ∅ ∧ ∃x ∈ Nn B = ((A +c x) +c 1c)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltfin 4441 | . 2 ⊢ <fin = {y ∣ ∃z∃w(y = ⟪z, w⟫ ∧ (z ≠ ∅ ∧ ∃x ∈ Nn w = ((z +c x) +c 1c)))} | |
2 | neeq1 2524 | . . 3 ⊢ (z = A → (z ≠ ∅ ↔ A ≠ ∅)) | |
3 | addceq1 4383 | . . . . . 6 ⊢ (z = A → (z +c x) = (A +c x)) | |
4 | 3 | addceq1d 4389 | . . . . 5 ⊢ (z = A → ((z +c x) +c 1c) = ((A +c x) +c 1c)) |
5 | 4 | eqeq2d 2364 | . . . 4 ⊢ (z = A → (w = ((z +c x) +c 1c) ↔ w = ((A +c x) +c 1c))) |
6 | 5 | rexbidv 2635 | . . 3 ⊢ (z = A → (∃x ∈ Nn w = ((z +c x) +c 1c) ↔ ∃x ∈ Nn w = ((A +c x) +c 1c))) |
7 | 2, 6 | anbi12d 691 | . 2 ⊢ (z = A → ((z ≠ ∅ ∧ ∃x ∈ Nn w = ((z +c x) +c 1c)) ↔ (A ≠ ∅ ∧ ∃x ∈ Nn w = ((A +c x) +c 1c)))) |
8 | eqeq1 2359 | . . . 4 ⊢ (w = B → (w = ((A +c x) +c 1c) ↔ B = ((A +c x) +c 1c))) | |
9 | 8 | rexbidv 2635 | . . 3 ⊢ (w = B → (∃x ∈ Nn w = ((A +c x) +c 1c) ↔ ∃x ∈ Nn B = ((A +c x) +c 1c))) |
10 | 9 | anbi2d 684 | . 2 ⊢ (w = B → ((A ≠ ∅ ∧ ∃x ∈ Nn w = ((A +c x) +c 1c)) ↔ (A ≠ ∅ ∧ ∃x ∈ Nn B = ((A +c x) +c 1c)))) |
11 | 1, 7, 10 | opkelopkabg 4245 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ <fin ↔ (A ≠ ∅ ∧ ∃x ∈ Nn B = ((A +c x) +c 1c)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 ∅c0 3550 ⟪copk 4057 1cc1c 4134 Nn cnnc 4373 +c cplc 4375 <fin cltfin 4433 |
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-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-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-rex 2620 df-v 2861 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-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-sik 4192 df-ssetk 4193 df-addc 4378 df-ltfin 4441 |
This theorem is referenced by: ltfinirr 4457 leltfintr 4458 ltfintr 4459 ltfinp1 4462 lefinlteq 4463 ltfintri 4466 ltlefin 4468 tfinltfinlem1 4500 tfinltfin 4501 sfinltfin 4535 |
Copyright terms: Public domain | W3C validator |