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Mirrors > Home > MPE Home > Th. List > Mathboxes > noxp1o | Structured version Visualization version GIF version |
Description: The Cartesian product of an ordinal and {1o} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.) |
Ref | Expression |
---|---|
noxp1o | ⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8109 | . . . . . 6 ⊢ 1o ∈ V | |
2 | 1 | prid1 4697 | . . . . 5 ⊢ 1o ∈ {1o, 2o} |
3 | 2 | fconst6 6568 | . . . 4 ⊢ (𝐴 × {1o}):𝐴⟶{1o, 2o} |
4 | 1 | snnz 4710 | . . . . . 6 ⊢ {1o} ≠ ∅ |
5 | dmxp 5798 | . . . . . 6 ⊢ ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ dom (𝐴 × {1o}) = 𝐴 |
7 | 6 | feq2i 6505 | . . . 4 ⊢ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o}) |
8 | 3, 7 | mpbir 233 | . . 3 ⊢ (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} |
9 | 8 | a1i 11 | . 2 ⊢ (𝐴 ∈ On → (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o}) |
10 | 6 | eleq1i 2903 | . . 3 ⊢ (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On) |
11 | 10 | biimpri 230 | . 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On) |
12 | elno3 33162 | . 2 ⊢ ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On)) | |
13 | 9, 11, 12 | sylanbrc 585 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 {csn 4566 {cpr 4568 × cxp 5552 dom cdm 5554 Oncon0 6190 ⟶wf 6350 1oc1o 8094 2oc2o 8095 No csur 33147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-1o 8101 df-no 33150 |
This theorem is referenced by: bdayfo 33182 |
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