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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 | 1on 7612 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
2 | 1 | elexi 3244 | . . . . . 6 ⊢ 1𝑜 ∈ V |
3 | 2 | prid1 4329 | . . . . 5 ⊢ 1𝑜 ∈ {1𝑜, 2𝑜} |
4 | 3 | fconst6 6133 | . . . 4 ⊢ (𝐴 × {1𝑜}):𝐴⟶{1𝑜, 2𝑜} |
5 | 2 | snnz 4340 | . . . . . 6 ⊢ {1𝑜} ≠ ∅ |
6 | dmxp 5376 | . . . . . 6 ⊢ ({1𝑜} ≠ ∅ → dom (𝐴 × {1𝑜}) = 𝐴) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ dom (𝐴 × {1𝑜}) = 𝐴 |
8 | 7 | feq2i 6075 | . . . 4 ⊢ ((𝐴 × {1𝑜}):dom (𝐴 × {1𝑜})⟶{1𝑜, 2𝑜} ↔ (𝐴 × {1𝑜}):𝐴⟶{1𝑜, 2𝑜}) |
9 | 4, 8 | mpbir 221 | . . 3 ⊢ (𝐴 × {1𝑜}):dom (𝐴 × {1𝑜})⟶{1𝑜, 2𝑜} |
10 | 9 | a1i 11 | . 2 ⊢ (𝐴 ∈ On → (𝐴 × {1𝑜}):dom (𝐴 × {1𝑜})⟶{1𝑜, 2𝑜}) |
11 | 7 | eleq1i 2721 | . . 3 ⊢ (dom (𝐴 × {1𝑜}) ∈ On ↔ 𝐴 ∈ On) |
12 | 11 | biimpri 218 | . 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1𝑜}) ∈ On) |
13 | elno3 31933 | . 2 ⊢ ((𝐴 × {1𝑜}) ∈ No ↔ ((𝐴 × {1𝑜}):dom (𝐴 × {1𝑜})⟶{1𝑜, 2𝑜} ∧ dom (𝐴 × {1𝑜}) ∈ On)) | |
14 | 10, 12, 13 | sylanbrc 699 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {1𝑜}) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∅c0 3948 {csn 4210 {cpr 4212 × cxp 5141 dom cdm 5143 Oncon0 5761 ⟶wf 5922 1𝑜c1o 7598 2𝑜c2o 7599 No csur 31918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-1o 7605 df-no 31921 |
This theorem is referenced by: bdayfo 31953 |
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