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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 31043 | . . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜}) | |
2 | onin 5671 | . . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∩ 𝐵) ∈ On) | |
3 | fresin 5986 | . . . . . . . 8 ⊢ (𝐴:𝑥⟶{1𝑜, 2𝑜} → (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1𝑜, 2𝑜}) | |
4 | feq2 5940 | . . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜} ↔ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1𝑜, 2𝑜})) | |
5 | 4 | rspcev 3282 | . . . . . . . 8 ⊢ (((𝑥 ∩ 𝐵) ∈ On ∧ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1𝑜, 2𝑜}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜}) |
6 | 2, 3, 5 | syl2an 493 | . . . . . . 7 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1𝑜, 2𝑜}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜}) |
7 | 6 | an32s 842 | . . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1𝑜, 2𝑜}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜}) |
8 | 7 | ex 449 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1𝑜, 2𝑜}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜})) |
9 | 8 | rexlimiva 3010 | . . . 4 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜})) |
10 | 9 | imp 444 | . . 3 ⊢ ((∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜}) |
11 | 1, 10 | sylanb 488 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜}) |
12 | elno 31043 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1𝑜, 2𝑜}) | |
13 | 11, 12 | sylibr 223 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 ∩ cin 3539 {cpr 4127 ↾ cres 5040 Oncon0 5640 ⟶wf 5800 1𝑜c1o 7440 2𝑜c2o 7441 No csur 31037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-no 31040 |
This theorem is referenced by: sltres 31061 |
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