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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 40293. (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 6010 | . . . 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 4164 | . . . 4 ⊢ (𝜑 → ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) ⊆ 𝑈) |
6 | 2, 5 | eqsstrid 4017 | . . 3 ⊢ (𝜑 → (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) |
7 | trclimalb2 40078 | . . 3 ⊢ ((𝑅 ∈ V ∧ (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) | |
8 | 1, 6, 7 | syl2anc 586 | . 2 ⊢ (𝜑 → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) |
9 | frege77d.ab | . . . 4 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | |
10 | df-br 5069 | . . . 4 ⊢ (𝐴(t+‘𝑅)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) | |
11 | 9, 10 | sylib 220 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) |
12 | frege77d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
13 | frege77d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
14 | elimasng 5957 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) | |
15 | 12, 13, 14 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) |
16 | 11, 15 | mpbird 259 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((t+‘𝑅) “ {𝐴})) |
17 | 8, 16 | sseldd 3970 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 {csn 4569 〈cop 4575 class class class wbr 5068 “ cima 5560 ‘cfv 6357 t+ctcl 14347 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-trcl 14349 df-relexp 14382 |
This theorem is referenced by: frege81d 40099 frege87d 40102 |
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