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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege130 | Structured version Visualization version GIF version |
Description: Lemma for frege131 39127. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege130.m | ⊢ 𝑀 ∈ 𝑈 |
frege130.r | ⊢ 𝑅 ∈ 𝑉 |
Ref | Expression |
---|---|
frege130 | ⊢ ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3417 | . . . . 5 ⊢ 𝑎 ∈ V | |
2 | vex 3417 | . . . . 5 ⊢ 𝑏 ∈ V | |
3 | frege130.m | . . . . 5 ⊢ 𝑀 ∈ 𝑈 | |
4 | frege130.r | . . . . 5 ⊢ 𝑅 ∈ 𝑉 | |
5 | 1, 2, 3, 4 | frege129 39125 | . . . 4 ⊢ (Fun ◡◡𝑅 → ((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))) |
6 | 5 | alrimdv 2028 | . . 3 ⊢ (Fun ◡◡𝑅 → ((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))) |
7 | 6 | alrimiv 2026 | . 2 ⊢ (Fun ◡◡𝑅 → ∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))) |
8 | frege9 38945 | . 2 ⊢ ((Fun ◡◡𝑅 → ∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))) → ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))) | |
9 | 7, 8 | ax-mp 5 | 1 ⊢ ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1654 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 {csn 4399 class class class wbr 4875 I cid 5251 ◡ccnv 5345 “ cima 5349 Fun wfun 6121 ‘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: frege131 39127 |
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