Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 8099 | . . . . . 6 ⊢ 1o ∈ V | |
2 | 1 | prid1 4690 | . . . . 5 ⊢ 1o ∈ {1o, 2o} |
3 | 2 | fconst6 6562 | . . . 4 ⊢ (𝐴 × {1o}):𝐴⟶{1o, 2o} |
4 | 1 | snnz 4703 | . . . . . 6 ⊢ {1o} ≠ ∅ |
5 | dmxp 5792 | . . . . . 6 ⊢ ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ dom (𝐴 × {1o}) = 𝐴 |
7 | 6 | feq2i 6499 | . . . 4 ⊢ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o}) |
8 | 3, 7 | mpbir 232 | . . 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 2900 | . . 3 ⊢ (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On) |
11 | 10 | biimpri 229 | . 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On) |
12 | elno3 33059 | . 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 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 {csn 4557 {cpr 4559 × cxp 5546 dom cdm 5548 Oncon0 6184 ⟶wf 6344 1oc1o 8084 2oc2o 8085 No csur 33044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1o 8091 df-no 33047 |
This theorem is referenced by: bdayfo 33079 |
Copyright terms: Public domain | W3C validator |