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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 40316 | . 2 ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
4 | frege110.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege110.m | . . . 4 ⊢ 𝑀 ∈ 𝐶 | |
6 | imaundir 6008 | . . . . 5 ⊢ (((t+‘𝑅) ∪ I ) “ {𝑋}) = (((t+‘𝑅) “ {𝑋}) ∪ ( I “ {𝑋})) | |
7 | fvex 6682 | . . . . . . 7 ⊢ (t+‘𝑅) ∈ V | |
8 | imaexg 7619 | . . . . . . 7 ⊢ ((t+‘𝑅) ∈ V → ((t+‘𝑅) “ {𝑋}) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ ((t+‘𝑅) “ {𝑋}) ∈ V |
10 | imai 5941 | . . . . . . 7 ⊢ ( I “ {𝑋}) = {𝑋} | |
11 | snex 5331 | . . . . . . 7 ⊢ {𝑋} ∈ V | |
12 | 10, 11 | eqeltri 2909 | . . . . . 6 ⊢ ( I “ {𝑋}) ∈ V |
13 | 9, 12 | unex 7468 | . . . . 5 ⊢ (((t+‘𝑅) “ {𝑋}) ∪ ( I “ {𝑋})) ∈ V |
14 | 6, 13 | eqeltri 2909 | . . . 4 ⊢ (((t+‘𝑅) ∪ I ) “ {𝑋}) ∈ V |
15 | 4, 5, 2, 14 | frege78 40285 | . . 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 3513 | . . . . . . 7 ⊢ 𝑋 ∈ V |
17 | vex 3497 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
18 | 16, 17 | elimasn 5953 | . . . . . 6 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑎〉 ∈ ((t+‘𝑅) ∪ I )) |
19 | df-br 5066 | . . . . . 6 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑎 ↔ 〈𝑋, 𝑎〉 ∈ ((t+‘𝑅) ∪ I )) | |
20 | 18, 19 | bitr4i 280 | . . . . 5 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 𝑋((t+‘𝑅) ∪ I )𝑎) |
21 | 20 | imbi2i 338 | . . . 4 ⊢ ((𝑌𝑅𝑎 → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) ↔ (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)) |
22 | 21 | albii 1816 | . . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)) |
23 | 5 | elexi 3513 | . . . . . 6 ⊢ 𝑀 ∈ V |
24 | 16, 23 | elimasn 5953 | . . . . 5 ⊢ (𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑀〉 ∈ ((t+‘𝑅) ∪ I )) |
25 | df-br 5066 | . . . . 5 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑀 ↔ 〈𝑋, 𝑀〉 ∈ ((t+‘𝑅) ∪ I )) | |
26 | 24, 25 | bitr4i 280 | . . . 4 ⊢ (𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 𝑋((t+‘𝑅) ∪ I )𝑀) |
27 | 26 | imbi2i 338 | . . 3 ⊢ ((𝑌(t+‘𝑅)𝑀 → 𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋})) ↔ (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) |
28 | 15, 22, 27 | 3imtr3g 297 | . 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 1531 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 {csn 4566 〈cop 4572 class class class wbr 5065 I cid 5458 “ cima 5557 ‘cfv 6354 t+ctcl 14344 hereditary whe 40116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-frege1 40134 ax-frege2 40135 ax-frege8 40153 ax-frege28 40174 ax-frege31 40178 ax-frege41 40189 ax-frege52a 40201 ax-frege52c 40232 ax-frege58b 40245 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-seq 13369 df-trcl 14346 df-relexp 14379 df-he 40117 |
This theorem is referenced by: frege124 40331 |
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