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Mirrors > Home > MPE Home > Th. List > Mathboxes > onno | Structured version Visualization version GIF version |
Description: Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.) |
Ref | Expression |
---|---|
onno | ⊢ (𝐴 ∈ On → (𝐴 × {2o}) ∈ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2oex 8529 | . . 3 ⊢ 2o ∈ V | |
2 | 1 | prid2 4788 | . 2 ⊢ 2o ∈ {1o, 2o} |
3 | onnog 43332 | . 2 ⊢ ((𝐴 ∈ On ∧ 2o ∈ {1o, 2o}) → (𝐴 × {2o}) ∈ No ) | |
4 | 2, 3 | mpan2 690 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {2o}) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 {csn 4648 {cpr 4650 × cxp 5697 Oncon0 6394 1oc1o 8511 2oc2o 8512 No csur 27693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-suc 6400 df-fun 6574 df-fn 6575 df-f 6576 df-1o 8518 df-2o 8519 df-no 27696 |
This theorem is referenced by: onnoi 43337 |
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