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Mirrors > Home > NFE Home > Th. List > elsuci | Unicode version |
Description: Lemma for ncfinraise 4481. Take a natural and a disjoint union and compute membership in the successor natural. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
elsuci.1 |
Ref | Expression |
---|---|
elsuci | 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci.1 | . . . . 5 | |
2 | 1 | elcompl 3225 | . . . 4 ∼ |
3 | eqid 2353 | . . . . 5 | |
4 | sneq 3744 | . . . . . . . 8 | |
5 | 4 | uneq2d 3418 | . . . . . . 7 |
6 | 5 | eqeq2d 2364 | . . . . . 6 |
7 | 6 | rspcev 2955 | . . . . 5 ∼ ∼ |
8 | 3, 7 | mpan2 652 | . . . 4 ∼ ∼ |
9 | 2, 8 | sylbir 204 | . . 3 ∼ |
10 | compleq 3243 | . . . . 5 ∼ ∼ | |
11 | uneq1 3411 | . . . . . 6 | |
12 | 11 | eqeq2d 2364 | . . . . 5 |
13 | 10, 12 | rexeqbidv 2820 | . . . 4 ∼ ∼ |
14 | 13 | rspcev 2955 | . . 3 ∼ ∼ |
15 | 9, 14 | sylan2 460 | . 2 ∼ |
16 | elsuc 4413 | . 2 1c ∼ | |
17 | 15, 16 | sylibr 203 | 1 1c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 358 wceq 1642 wcel 1710 wrex 2615 cvv 2859 ∼ ccompl 3205 cun 3207 csn 3737 1cc1c 4134 cplc 4375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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: prepeano4 4451 ncfinraise 4481 ncfinlower 4483 tfinsuc 4498 nnadjoin 4520 sfindbl 4530 tfinnn 4534 nulnnn 4556 |
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