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Mirrors > Home > ILE Home > Th. List > phplem3g | Unicode version |
Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6800 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
phplem3g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2220 | . . . . 5 | |
2 | 1 | anbi2d 460 | . . . 4 |
3 | sneq 3571 | . . . . . 6 | |
4 | 3 | difeq2d 3225 | . . . . 5 |
5 | 4 | breq2d 3978 | . . . 4 |
6 | 2, 5 | imbi12d 233 | . . 3 |
7 | eleq1 2220 | . . . . . . 7 | |
8 | suceq 4363 | . . . . . . . 8 | |
9 | 8 | eleq2d 2227 | . . . . . . 7 |
10 | 7, 9 | anbi12d 465 | . . . . . 6 |
11 | id 19 | . . . . . . 7 | |
12 | 8 | difeq1d 3224 | . . . . . . 7 |
13 | 11, 12 | breq12d 3979 | . . . . . 6 |
14 | 10, 13 | imbi12d 233 | . . . . 5 |
15 | vex 2715 | . . . . . 6 | |
16 | vex 2715 | . . . . . 6 | |
17 | 15, 16 | phplem3 6800 | . . . . 5 |
18 | 14, 17 | vtoclg 2772 | . . . 4 |
19 | 18 | anabsi5 569 | . . 3 |
20 | 6, 19 | vtoclg 2772 | . 2 |
21 | 20 | anabsi7 571 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cdif 3099 csn 3560 class class class wbr 3966 csuc 4326 com 4550 cen 6684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-en 6687 |
This theorem is referenced by: phplem4dom 6808 phpm 6811 phplem4on 6813 |
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