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Mirrors > Home > NFE Home > Th. List > nnceleq | Unicode version |
Description: If two naturals have an element in common, then they are equal. (Contributed by SF, 13-Feb-2015.) |
Ref | Expression |
---|---|
nnceleq | Nn Nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3219 | . . . 4 | |
2 | n0i 3555 | . . . 4 | |
3 | 1, 2 | sylbir 204 | . . 3 |
4 | 3 | adantl 452 | . 2 Nn Nn |
5 | nndisjeq 4429 | . . 3 Nn Nn | |
6 | 5 | adantr 451 | . 2 Nn Nn |
7 | orel1 371 | . 2 | |
8 | 4, 6, 7 | sylc 56 | 1 Nn Nn |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 357 wa 358 wceq 1642 wcel 1710 cin 3208 c0 3550 Nn cnnc 4373 |
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-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 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-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-0c 4377 df-addc 4378 df-nnc 4379 |
This theorem is referenced by: prepeano4 4451 vfinnc 4471 ncfindi 4475 ncfinsn 4476 ncfineleq 4477 nnpw1ex 4484 tfin11 4493 tfinpw1 4494 ncfintfin 4495 tfindi 4496 tfin0c 4497 tfinsuc 4498 sfin112 4529 vfintle 4546 vfinspsslem1 4550 vfinncsp 4554 |
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