| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > naddfn | Structured version Visualization version GIF version | ||
| Description: Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.) |
| Ref | Expression |
|---|---|
| naddfn | ⊢ +no Fn (On × On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-nadd 8704 | . 2 ⊢ +no = frecs({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ ∩ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st ‘𝑧)} × (2nd ‘𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st ‘𝑧) × {(2nd ‘𝑧)})) ⊆ 𝑤)})) | |
| 2 | 1 | on2recsfn 8705 | 1 ⊢ +no Fn (On × On) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 {crab 3436 Vcvv 3480 ⊆ wss 3951 {csn 4626 ∩ cint 4946 × cxp 5683 “ cima 5688 Oncon0 6384 Fn wfn 6556 ‘cfv 6561 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 +no cnadd 8703 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-frecs 8306 df-nadd 8704 |
| This theorem is referenced by: naddcllem 8714 naddov2 8717 naddf 8719 naddunif 8731 naddasslem1 8732 naddasslem2 8733 |
| Copyright terms: Public domain | W3C validator |