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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 40164. (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 6001 | . . . 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 4159 | . . . 4 ⊢ (𝜑 → ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) ⊆ 𝑈) |
6 | 2, 5 | eqsstrid 4012 | . . 3 ⊢ (𝜑 → (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) |
7 | trclimalb2 39949 | . . 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 5058 | . . . 4 ⊢ (𝐴(t+‘𝑅)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) | |
11 | 9, 10 | sylib 219 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) |
12 | frege77d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
13 | frege77d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
14 | elimasng 5948 | . . . 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 258 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((t+‘𝑅) “ {𝐴})) |
17 | 8, 16 | sseldd 3965 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 {csn 4557 〈cop 4563 class class class wbr 5057 “ cima 5551 ‘cfv 6348 t+ctcl 14333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 df-trcl 14335 df-relexp 14368 |
This theorem is referenced by: frege81d 39970 frege87d 39973 |
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