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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege110 | Structured version Visualization version GIF version |
Description: Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege110.x | ⊢ 𝑋 ∈ 𝐴 |
frege110.y | ⊢ 𝑌 ∈ 𝐵 |
frege110.m | ⊢ 𝑀 ∈ 𝐶 |
frege110.r | ⊢ 𝑅 ∈ 𝐷 |
Ref | Expression |
---|---|
frege110 | ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege110.x | . . 3 ⊢ 𝑋 ∈ 𝐴 | |
2 | frege110.r | . . 3 ⊢ 𝑅 ∈ 𝐷 | |
3 | 1, 2 | frege109 39105 | . 2 ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
4 | frege110.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege110.m | . . . 4 ⊢ 𝑀 ∈ 𝐶 | |
6 | imaundir 5791 | . . . . 5 ⊢ (((t+‘𝑅) ∪ I ) “ {𝑋}) = (((t+‘𝑅) “ {𝑋}) ∪ ( I “ {𝑋})) | |
7 | fvex 6450 | . . . . . . 7 ⊢ (t+‘𝑅) ∈ V | |
8 | imaexg 7370 | . . . . . . 7 ⊢ ((t+‘𝑅) ∈ V → ((t+‘𝑅) “ {𝑋}) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ ((t+‘𝑅) “ {𝑋}) ∈ V |
10 | imai 5723 | . . . . . . 7 ⊢ ( I “ {𝑋}) = {𝑋} | |
11 | snex 5131 | . . . . . . 7 ⊢ {𝑋} ∈ V | |
12 | 10, 11 | eqeltri 2902 | . . . . . 6 ⊢ ( I “ {𝑋}) ∈ V |
13 | 9, 12 | unex 7221 | . . . . 5 ⊢ (((t+‘𝑅) “ {𝑋}) ∪ ( I “ {𝑋})) ∈ V |
14 | 6, 13 | eqeltri 2902 | . . . 4 ⊢ (((t+‘𝑅) ∪ I ) “ {𝑋}) ∈ V |
15 | 4, 5, 2, 14 | frege78 39074 | . . 3 ⊢ (𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) → (∀𝑎(𝑌𝑅𝑎 → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) → (𝑌(t+‘𝑅)𝑀 → 𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})))) |
16 | 1 | elexi 3430 | . . . . . . 7 ⊢ 𝑋 ∈ V |
17 | vex 3417 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
18 | 16, 17 | elimasn 5735 | . . . . . 6 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑎〉 ∈ ((t+‘𝑅) ∪ I )) |
19 | df-br 4876 | . . . . . 6 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑎 ↔ 〈𝑋, 𝑎〉 ∈ ((t+‘𝑅) ∪ I )) | |
20 | 18, 19 | bitr4i 270 | . . . . 5 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 𝑋((t+‘𝑅) ∪ I )𝑎) |
21 | 20 | imbi2i 328 | . . . 4 ⊢ ((𝑌𝑅𝑎 → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) ↔ (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)) |
22 | 21 | albii 1918 | . . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)) |
23 | 5 | elexi 3430 | . . . . . 6 ⊢ 𝑀 ∈ V |
24 | 16, 23 | elimasn 5735 | . . . . 5 ⊢ (𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑀〉 ∈ ((t+‘𝑅) ∪ I )) |
25 | df-br 4876 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑀 ↔ 〈𝑋, 𝑀〉 ∈ ((t+‘𝑅) ∪ I )) | |
26 | 24, 25 | bitr4i 270 | . . . 4 ⊢ (𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 𝑋((t+‘𝑅) ∪ I )𝑀) |
27 | 26 | imbi2i 328 | . . 3 ⊢ ((𝑌(t+‘𝑅)𝑀 → 𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) ↔ (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) |
28 | 15, 22, 27 | 3imtr3g 287 | . 2 ⊢ (𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) → (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))) |
29 | 3, 28 | ax-mp 5 | 1 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1654 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 {csn 4399 〈cop 4405 class class class wbr 4875 I cid 5251 “ cima 5349 ‘cfv 6127 t+ctcl 14110 hereditary whe 38905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-frege1 38923 ax-frege2 38924 ax-frege8 38942 ax-frege28 38963 ax-frege31 38967 ax-frege41 38978 ax-frege52a 38990 ax-frege52c 39021 ax-frege58b 39034 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ifp 1090 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-seq 13103 df-trcl 14112 df-relexp 14145 df-he 38906 |
This theorem is referenced by: frege124 39120 |
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