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Mirrors > Home > NFE Home > Th. List > elsuc | Unicode version |
Description: Membership in a successor. Theorem X.1.16 of [Rosser] p. 279. (Contributed by SF, 16-Jan-2015.) |
Ref | Expression |
---|---|
elsuc | 1c ∼ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eladdc 4398 | . 2 1c 1c | |
2 | snex 4111 | . . . . . . 7 | |
3 | ineq2 3451 | . . . . . . . . 9 | |
4 | 3 | eqeq1d 2361 | . . . . . . . 8 |
5 | uneq2 3412 | . . . . . . . . 9 | |
6 | 5 | eqeq2d 2364 | . . . . . . . 8 |
7 | 4, 6 | anbi12d 691 | . . . . . . 7 |
8 | 2, 7 | ceqsexv 2894 | . . . . . 6 |
9 | disjsn 3786 | . . . . . . . 8 | |
10 | vex 2862 | . . . . . . . . 9 | |
11 | 10 | elcompl 3225 | . . . . . . . 8 ∼ |
12 | 9, 11 | bitr4i 243 | . . . . . . 7 ∼ |
13 | 12 | anbi1i 676 | . . . . . 6 ∼ |
14 | 8, 13 | bitri 240 | . . . . 5 ∼ |
15 | 14 | exbii 1582 | . . . 4 ∼ |
16 | df-rex 2620 | . . . . 5 1c 1c | |
17 | el1c 4139 | . . . . . . . . 9 1c | |
18 | 17 | anbi1i 676 | . . . . . . . 8 1c |
19 | 19.41v 1901 | . . . . . . . 8 | |
20 | 18, 19 | bitr4i 243 | . . . . . . 7 1c |
21 | 20 | exbii 1582 | . . . . . 6 1c |
22 | excom 1741 | . . . . . 6 | |
23 | 21, 22 | bitri 240 | . . . . 5 1c |
24 | 16, 23 | bitri 240 | . . . 4 1c |
25 | df-rex 2620 | . . . 4 ∼ ∼ | |
26 | 15, 24, 25 | 3bitr4i 268 | . . 3 1c ∼ |
27 | 26 | rexbii 2639 | . 2 1c ∼ |
28 | 1, 27 | bitri 240 | 1 1c ∼ |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 wrex 2615 ∼ ccompl 3205 cun 3207 cin 3208 c0 3550 csn 3737 1cc1c 4134 cplc 4375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-1c 4136 df-addc 4378 |
This theorem is referenced by: elsuci 4414 nnsucelr 4428 nndisjeq 4429 prepeano4 4451 ncfinraise 4481 ncfinlower 4483 tfinsuc 4498 oddfinex 4504 nnadjoin 4520 nnpweq 4523 sfindbl 4530 tfinnn 4534 peano4nc 6150 el2c 6191 nmembers1lem3 6270 |
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