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Mirrors > Home > NFE Home > Th. List > opklefing | GIF version |
Description: Kuratowski ordered pair membership in finite less than or equal to. (Contributed by SF, 18-Jan-2015.) |
Ref | Expression |
---|---|
opklefing | ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ ≤fin ↔ ∃x ∈ Nn B = (A +c x))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lefin 4440 | . 2 ⊢ ≤fin = {w ∣ ∃y∃z(w = ⟪y, z⟫ ∧ ∃x ∈ Nn z = (y +c x))} | |
2 | addceq1 4383 | . . . 4 ⊢ (y = A → (y +c x) = (A +c x)) | |
3 | 2 | eqeq2d 2364 | . . 3 ⊢ (y = A → (z = (y +c x) ↔ z = (A +c x))) |
4 | 3 | rexbidv 2635 | . 2 ⊢ (y = A → (∃x ∈ Nn z = (y +c x) ↔ ∃x ∈ Nn z = (A +c x))) |
5 | eqeq1 2359 | . . 3 ⊢ (z = B → (z = (A +c x) ↔ B = (A +c x))) | |
6 | 5 | rexbidv 2635 | . 2 ⊢ (z = B → (∃x ∈ Nn z = (A +c x) ↔ ∃x ∈ Nn B = (A +c x))) |
7 | 1, 4, 6 | opkelopkabg 4245 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ ≤fin ↔ ∃x ∈ Nn B = (A +c x))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ⟪copk 4057 Nn cnnc 4373 +c cplc 4375 ≤fin clefin 4432 |
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-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-lefin 4440 |
This theorem is referenced by: lefinaddc 4450 nulge 4456 leltfintr 4458 lefinlteq 4463 ltfintri 4466 lefinrflx 4467 ltlefin 4468 vfinspsslem1 4550 |
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