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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 8193 | . . . . . 6 ⊢ 1o ∈ V | |
2 | 1 | prid1 4664 | . . . . 5 ⊢ 1o ∈ {1o, 2o} |
3 | 2 | fconst6 6587 | . . . 4 ⊢ (𝐴 × {1o}):𝐴⟶{1o, 2o} |
4 | 1 | snnz 4678 | . . . . . 6 ⊢ {1o} ≠ ∅ |
5 | dmxp 5783 | . . . . . 6 ⊢ ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ dom (𝐴 × {1o}) = 𝐴 |
7 | 6 | feq2i 6515 | . . . 4 ⊢ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o}) |
8 | 3, 7 | mpbir 234 | . . 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 2821 | . . 3 ⊢ (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On) |
11 | 10 | biimpri 231 | . 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On) |
12 | elno3 33544 | . 2 ⊢ ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On)) | |
13 | 9, 11, 12 | sylanbrc 586 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 {csn 4527 {cpr 4529 × cxp 5534 dom cdm 5536 Oncon0 6191 ⟶wf 6354 1oc1o 8173 2oc2o 8174 No csur 33529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1o 8180 df-no 33532 |
This theorem is referenced by: bdayfo 33566 |
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