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Mirrors > Home > MPE Home > Th. List > Mathboxes > noxpsgn | Structured version Visualization version GIF version |
Description: The Cartesian product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.) |
Ref | Expression |
---|---|
noxpsgn.1 | ⊢ 𝑋 ∈ {1𝑜, 2𝑜} |
Ref | Expression |
---|---|
noxpsgn | ⊢ (𝐴 ∈ On → (𝐴 × {𝑋}) ∈ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noxpsgn.1 | . . . 4 ⊢ 𝑋 ∈ {1𝑜, 2𝑜} | |
2 | 1 | fconst6 6008 | . . 3 ⊢ (𝐴 × {𝑋}):𝐴⟶{1𝑜, 2𝑜} |
3 | feq2 5940 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜} ↔ (𝐴 × {𝑋}):𝐴⟶{1𝑜, 2𝑜})) | |
4 | 3 | rspcev 3282 | . . 3 ⊢ ((𝐴 ∈ On ∧ (𝐴 × {𝑋}):𝐴⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On (𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜}) |
5 | 2, 4 | mpan2 703 | . 2 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜}) |
6 | elno 31043 | . 2 ⊢ ((𝐴 × {𝑋}) ∈ No ↔ ∃𝑥 ∈ On (𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜}) | |
7 | 5, 6 | sylibr 223 | 1 ⊢ (𝐴 ∈ On → (𝐴 × {𝑋}) ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∃wrex 2897 {csn 4125 {cpr 4127 × cxp 5036 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-id 4953 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-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: noxp1o 31063 noxp2o 31064 nobndlem3 31093 |
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