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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 8426 | . . . . . 6 ⊢ 1o ∈ V | |
2 | 1 | prid1 4727 | . . . . 5 ⊢ 1o ∈ {1o, 2o} |
3 | 2 | fconst6 6736 | . . . 4 ⊢ (𝐴 × {1o}):𝐴⟶{1o, 2o} |
4 | 1 | snnz 4741 | . . . . . 6 ⊢ {1o} ≠ ∅ |
5 | dmxp 5888 | . . . . . 6 ⊢ ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ dom (𝐴 × {1o}) = 𝐴 |
7 | 6 | feq2i 6664 | . . . 4 ⊢ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o}) |
8 | 3, 7 | mpbir 230 | . . 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 2825 | . . 3 ⊢ (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On) |
11 | 10 | biimpri 227 | . 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On) |
12 | elno3 27026 | . 2 ⊢ ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On)) | |
13 | 9, 11, 12 | sylanbrc 584 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∅c0 4286 {csn 4590 {cpr 4592 × cxp 5635 dom cdm 5637 Oncon0 6321 ⟶wf 6496 1oc1o 8409 2oc2o 8410 No csur 27011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-1o 8416 df-no 27014 |
This theorem is referenced by: bdayfo 27048 |
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