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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpg | Structured version Visualization version GIF version |
Description: Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿 ∣ 𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.) |
Ref | Expression |
---|---|
elpg | ⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpglem1 46085 | . . . 4 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) | |
2 | elpglem2 46086 | . . . 4 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) | |
3 | 1, 2 | impbii 212 | . . 3 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
4 | 3 | anbi2i 626 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))) |
5 | df-pg 46084 | . . . 4 ⊢ Pg = setrecs((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))) | |
6 | 5 | elsetrecs 46074 | . . 3 ⊢ (𝐴 ∈ Pg ↔ ∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥))) |
7 | elpglem3 46087 | . . 3 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))) | |
8 | 6, 7 | bitri 278 | . 2 ⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))) |
9 | 3anass 1097 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))) | |
10 | 4, 8, 9 | 3bitr4i 306 | 1 ⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∃wex 1787 ∈ wcel 2110 Vcvv 3405 ⊆ wss 3863 𝒫 cpw 4510 ↦ cmpt 5132 × cxp 5546 ‘cfv 6377 1st c1st 7756 2nd c2nd 7757 Pgcpg 46083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-reg 9205 ax-inf2 9253 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-r1 9377 df-rank 9378 df-setrecs 46059 df-pg 46084 |
This theorem is referenced by: (None) |
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