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Mirrors > Home > MPE Home > Th. List > nofv | Structured version Visualization version GIF version |
Description: The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.) |
Ref | Expression |
---|---|
nofv | ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1 896 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴) | |
2 | ndmfv 6942 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)) |
4 | nofun 27709 | . . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴) | |
5 | norn 27711 | . . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o}) | |
6 | fvelrn 7096 | . . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴) | |
7 | ssel 3989 | . . . . . . . 8 ⊢ (ran 𝐴 ⊆ {1o, 2o} → ((𝐴‘𝑋) ∈ ran 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o})) | |
8 | 6, 7 | syl5com 31 | . . . . . . 7 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (ran 𝐴 ⊆ {1o, 2o} → (𝐴‘𝑋) ∈ {1o, 2o})) |
9 | 8 | impancom 451 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o})) |
10 | 1oex 8515 | . . . . . . 7 ⊢ 1o ∈ V | |
11 | 2on 8519 | . . . . . . . 8 ⊢ 2o ∈ On | |
12 | 11 | elexi 3501 | . . . . . . 7 ⊢ 2o ∈ V |
13 | 10, 12 | elpr2 4657 | . . . . . 6 ⊢ ((𝐴‘𝑋) ∈ {1o, 2o} ↔ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)) |
14 | 9, 13 | imbitrdi 251 | . . . . 5 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))) |
15 | 4, 5, 14 | syl2anc 584 | . . . 4 ⊢ (𝐴 ∈ No → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))) |
16 | 3, 15 | orim12d 966 | . . 3 ⊢ (𝐴 ∈ No → ((¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴) → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))) |
17 | 1, 16 | mpi 20 | . 2 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))) |
18 | 3orass 1089 | . 2 ⊢ (((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o) ↔ ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))) | |
19 | 17, 18 | sylibr 234 | 1 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 {cpr 4633 dom cdm 5689 ran crn 5690 Oncon0 6386 Fun wfun 6557 ‘cfv 6563 1oc1o 8498 2oc2o 8499 No csur 27699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1o 8505 df-2o 8506 df-no 27702 |
This theorem is referenced by: nolesgn2o 27731 nogesgn1o 27733 nosep1o 27741 nosep2o 27742 nolt02o 27755 nogt01o 27756 nosupbnd1lem5 27772 nosupbnd1lem6 27773 noinfbnd1lem5 27787 noinfbnd1lem6 27788 |
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