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| Mirrors > Home > MPE Home > Th. List > 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 8517 | . . . . . 6 ⊢ 1o ∈ V | |
| 2 | 1 | prid1 4761 | . . . . 5 ⊢ 1o ∈ {1o, 2o} | 
| 3 | 2 | fconst6 6797 | . . . 4 ⊢ (𝐴 × {1o}):𝐴⟶{1o, 2o} | 
| 4 | 1 | snnz 4775 | . . . . . 6 ⊢ {1o} ≠ ∅ | 
| 5 | dmxp 5938 | . . . . . 6 ⊢ ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴) | |
| 6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ dom (𝐴 × {1o}) = 𝐴 | 
| 7 | 6 | feq2i 6727 | . . . 4 ⊢ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o}) | 
| 8 | 3, 7 | mpbir 231 | . . 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 2831 | . . 3 ⊢ (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On) | 
| 11 | 10 | biimpri 228 | . 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On) | 
| 12 | elno3 27701 | . 2 ⊢ ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On)) | |
| 13 | 9, 11, 12 | sylanbrc 583 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 {csn 4625 {cpr 4627 × cxp 5682 dom cdm 5684 Oncon0 6383 ⟶wf 6556 1oc1o 8500 2oc2o 8501 No csur 27685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-suc 6389 df-fun 6562 df-fn 6563 df-f 6564 df-1o 8507 df-no 27688 | 
| This theorem is referenced by: bdayfo 27723 | 
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