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Mirrors > Home > MPE Home > Th. List > elno3 | Structured version Visualization version GIF version |
Description: Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.) |
Ref | Expression |
---|---|
elno3 | ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anan32 1098 | . 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On)) | |
2 | elno2 27025 | . 2 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o})) | |
3 | df-f 6504 | . . . 4 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o})) | |
4 | funfn 6535 | . . . . 5 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | |
5 | 4 | anbi1i 625 | . . . 4 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o})) |
6 | 3, 5 | bitr4i 278 | . . 3 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o})) |
7 | 6 | anbi1i 625 | . 2 ⊢ ((𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On)) |
8 | 1, 2, 7 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3914 {cpr 4592 dom cdm 5637 ran crn 5638 Oncon0 6321 Fun wfun 6494 Fn wfn 6495 ⟶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-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-no 27014 |
This theorem is referenced by: noxp1o 27034 noseponlem 27035 |
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