Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 40338 | . 2 ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) |
4 | frege110.y | . . . 4 ⊢ 𝑌 ∈ 𝐵 | |
5 | frege110.m | . . . 4 ⊢ 𝑀 ∈ 𝐶 | |
6 | imaundir 6009 | . . . . 5 ⊢ (((t+‘𝑅) ∪ I ) “ {𝑋}) = (((t+‘𝑅) “ {𝑋}) ∪ ( I “ {𝑋})) | |
7 | fvex 6683 | . . . . . . 7 ⊢ (t+‘𝑅) ∈ V | |
8 | imaexg 7620 | . . . . . . 7 ⊢ ((t+‘𝑅) ∈ V → ((t+‘𝑅) “ {𝑋}) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ ((t+‘𝑅) “ {𝑋}) ∈ V |
10 | imai 5942 | . . . . . . 7 ⊢ ( I “ {𝑋}) = {𝑋} | |
11 | snex 5332 | . . . . . . 7 ⊢ {𝑋} ∈ V | |
12 | 10, 11 | eqeltri 2909 | . . . . . 6 ⊢ ( I “ {𝑋}) ∈ V |
13 | 9, 12 | unex 7469 | . . . . 5 ⊢ (((t+‘𝑅) “ {𝑋}) ∪ ( I “ {𝑋})) ∈ V |
14 | 6, 13 | eqeltri 2909 | . . . 4 ⊢ (((t+‘𝑅) ∪ I ) “ {𝑋}) ∈ V |
15 | 4, 5, 2, 14 | frege78 40307 | . . 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 5954 | . . . . . 6 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑎〉 ∈ ((t+‘𝑅) ∪ I )) |
19 | df-br 5067 | . . . . . 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 1820 | . . 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 5954 | . . . . 5 ⊢ (𝑀 ∈ (((t+‘𝑅) ∪ I ) “ {𝑋}) ↔ 〈𝑋, 𝑀〉 ∈ ((t+‘𝑅) ∪ I )) |
25 | df-br 5067 | . . . . 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 1535 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 {csn 4567 〈cop 4573 class class class wbr 5066 I cid 5459 “ cima 5558 ‘cfv 6355 t+ctcl 14345 hereditary whe 40138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-frege1 40156 ax-frege2 40157 ax-frege8 40175 ax-frege28 40196 ax-frege31 40200 ax-frege41 40211 ax-frege52a 40223 ax-frege52c 40254 ax-frege58b 40267 |
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 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-trcl 14347 df-relexp 14380 df-he 40139 |
This theorem is referenced by: frege124 40353 |
Copyright terms: Public domain | W3C validator |