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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 5279. (Revised by SN, 5-Jun-2025.) | 
| Ref | Expression | 
|---|---|
| elno | ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ No → 𝐴 ∈ V) | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | prex 5437 | . . . 4 ⊢ {1o, 2o} ∈ V | |
| 4 | fex2 7958 | . . . 4 ⊢ ((𝐴:𝑥⟶{1o, 2o} ∧ 𝑥 ∈ V ∧ {1o, 2o} ∈ V) → 𝐴 ∈ V) | |
| 5 | 2, 3, 4 | mp3an23 1455 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V) | 
| 6 | 5 | rexlimivw 3151 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → 𝐴 ∈ V) | 
| 7 | feq1 6716 | . . . 4 ⊢ (𝑓 = 𝐴 → (𝑓:𝑥⟶{1o, 2o} ↔ 𝐴:𝑥⟶{1o, 2o})) | |
| 8 | 7 | rexbidv 3179 | . . 3 ⊢ (𝑓 = 𝐴 → (∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o} ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})) | 
| 9 | df-no 27687 | . . 3 ⊢ No = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶{1o, 2o}} | |
| 10 | 8, 9 | elab2g 3680 | . 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 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 {cpr 4628 Oncon0 6384 ⟶wf 6557 1oc1o 8499 2oc2o 8500 No csur 27684 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-no 27687 | 
| This theorem is referenced by: nofun 27694 nodmon 27695 norn 27696 elno2 27699 noreson 27705 | 
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