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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 8662 | . 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 8663 | 1 ⊢ +no Fn (On × On) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 {crab 3424 Vcvv 3466 ⊆ wss 3941 {csn 4621 ∩ cint 4941 × cxp 5665 “ cima 5670 Oncon0 6355 Fn wfn 6529 ‘cfv 6534 ∈ cmpo 7404 1st c1st 7967 2nd c2nd 7968 +no cnadd 8661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-1st 7969 df-2nd 7970 df-frecs 8262 df-nadd 8662 |
This theorem is referenced by: naddcllem 8672 naddov2 8675 naddf 8677 naddunif 8689 naddasslem1 8690 naddasslem2 8691 |
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