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| Mirrors > Home > MPE Home > Th. List > noreson | Structured version Visualization version GIF version | ||
| Description: The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.) |
| Ref | Expression |
|---|---|
| noreson | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elno 27690 | . . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
| 2 | onin 6415 | . . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∩ 𝐵) ∈ On) | |
| 3 | fresin 6777 | . . . . . . . 8 ⊢ (𝐴:𝑥⟶{1o, 2o} → (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}) | |
| 4 | feq2 6717 | . . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ↾ 𝐵):𝑦⟶{1o, 2o} ↔ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o})) | |
| 5 | 4 | rspcev 3622 | . . . . . . . 8 ⊢ (((𝑥 ∩ 𝐵) ∈ On ∧ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}) |
| 6 | 2, 3, 5 | syl2an 596 | . . . . . . 7 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}) |
| 7 | 6 | an32s 652 | . . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}) |
| 8 | 7 | ex 412 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})) |
| 9 | 8 | rexlimiva 3147 | . . . 4 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})) |
| 10 | 9 | imp 406 | . . 3 ⊢ ((∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}) |
| 11 | 1, 10 | sylanb 581 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}) |
| 12 | elno 27690 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}) | |
| 13 | 11, 12 | sylibr 234 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 ∩ cin 3950 {cpr 4628 ↾ cres 5687 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-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ord 6387 df-on 6388 df-fun 6563 df-fn 6564 df-f 6565 df-no 27687 |
| This theorem is referenced by: sltres 27707 nodenselem6 27734 noresle 27742 nosupbnd1lem1 27753 nosupbnd1lem2 27754 nosupbnd1lem6 27758 nosupbnd1 27759 nosupbnd2lem1 27760 nosupbnd2 27761 noinfbnd1lem1 27768 noinfbnd1lem2 27769 noinfbnd1lem6 27773 noinfbnd1 27774 noinfbnd2lem1 27775 noinfbnd2 27776 nosupinfsep 27777 noetasuplem4 27781 noetainflem4 27785 |
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