Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6832 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
phplem3g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2233 | . . . . 5 | |
2 | 1 | anbi2d 461 | . . . 4 |
3 | sneq 3594 | . . . . . 6 | |
4 | 3 | difeq2d 3245 | . . . . 5 |
5 | 4 | breq2d 4001 | . . . 4 |
6 | 2, 5 | imbi12d 233 | . . 3 |
7 | eleq1 2233 | . . . . . . 7 | |
8 | suceq 4387 | . . . . . . . 8 | |
9 | 8 | eleq2d 2240 | . . . . . . 7 |
10 | 7, 9 | anbi12d 470 | . . . . . 6 |
11 | id 19 | . . . . . . 7 | |
12 | 8 | difeq1d 3244 | . . . . . . 7 |
13 | 11, 12 | breq12d 4002 | . . . . . 6 |
14 | 10, 13 | imbi12d 233 | . . . . 5 |
15 | vex 2733 | . . . . . 6 | |
16 | vex 2733 | . . . . . 6 | |
17 | 15, 16 | phplem3 6832 | . . . . 5 |
18 | 14, 17 | vtoclg 2790 | . . . 4 |
19 | 18 | anabsi5 574 | . . 3 |
20 | 6, 19 | vtoclg 2790 | . 2 |
21 | 20 | anabsi7 576 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cdif 3118 csn 3583 class class class wbr 3989 csuc 4350 com 4574 cen 6716 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-en 6719 |
This theorem is referenced by: phplem4dom 6840 phpm 6843 phplem4on 6845 |
Copyright terms: Public domain | W3C validator |