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Mirrors > Home > MPE Home > Th. List > elno | Structured version Visualization version GIF version |
Description: Membership in the surreals. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) Avoid ax-rep 5285. (Revised by SN, 5-Jun-2025.) |
Ref | Expression |
---|---|
elno | ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3499 | . 2 ⊢ (𝐴 ∈ No → 𝐴 ∈ V) | |
2 | vex 3482 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | prex 5443 | . . . 4 ⊢ {1o, 2o} ∈ V | |
4 | fex2 7957 | . . . 4 ⊢ ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ V ∧ {1o, 2o} ∈ V) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | mp3an23 1452 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V) |
6 | 5 | rexlimivw 3149 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V) |
7 | feq1 6717 | . . . 4 ⊢ (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o})) | |
8 | 7 | rexbidv 3177 | . . 3 ⊢ (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})) |
9 | df-no 27702 | . . 3 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} | |
10 | 8, 9 | elab2g 3683 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})) |
11 | 1, 6, 10 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 {cpr 4633 Oncon0 6386 ⟶wf 6559 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-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-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 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-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-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-no 27702 |
This theorem is referenced by: nofun 27709 nodmon 27710 norn 27711 elno2 27714 noreson 27720 |
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