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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege77d | Structured version Visualization version GIF version |
Description: If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 41529. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege77d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege77d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
frege77d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege77d.ab | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
frege77d.he | ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) |
frege77d.ss | ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) |
Ref | Expression |
---|---|
frege77d | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege77d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | imaundi 6046 | . . . 4 ⊢ (𝑅 “ ({𝐴} ∪ 𝑈)) = ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) | |
3 | frege77d.ss | . . . . 5 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) | |
4 | frege77d.he | . . . . 5 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) | |
5 | 3, 4 | unssd 4119 | . . . 4 ⊢ (𝜑 → ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) ⊆ 𝑈) |
6 | 2, 5 | eqsstrid 3968 | . . 3 ⊢ (𝜑 → (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) |
7 | trclimalb2 41315 | . . 3 ⊢ ((𝑅 ∈ V ∧ (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) | |
8 | 1, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) |
9 | frege77d.ab | . . . 4 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | |
10 | df-br 5074 | . . . 4 ⊢ (𝐴(t+‘𝑅)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) | |
11 | 9, 10 | sylib 217 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) |
12 | frege77d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
13 | frege77d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
14 | elimasng 5989 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) | |
15 | 12, 13, 14 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) |
16 | 11, 15 | mpbird 256 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((t+‘𝑅) “ {𝐴})) |
17 | 8, 16 | sseldd 3921 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 Vcvv 3429 ∪ cun 3884 ⊆ wss 3886 {csn 4561 〈cop 4567 class class class wbr 5073 “ cima 5587 ‘cfv 6426 t+ctcl 14706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-n0 12244 df-z 12330 df-uz 12593 df-seq 13732 df-trcl 14708 df-relexp 14741 |
This theorem is referenced by: frege81d 41336 frege87d 41339 |
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