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Mirrors > Home > NFE Home > Th. List > prepeano4 | Unicode version |
Description: Assuming a non-null successor, cardinal successor is one-to-one. Theorem X.1.19 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.) |
Ref | Expression |
---|---|
prepeano4 | Nn Nn 1c 1c 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3559 | . . 3 1c 1c | |
2 | elsuc 4413 | . . . . 5 1c ∼ | |
3 | simplll 734 | . . . . . . . 8 Nn Nn 1c 1c ∼ Nn | |
4 | simpllr 735 | . . . . . . . 8 Nn Nn 1c 1c ∼ Nn | |
5 | simprl 732 | . . . . . . . 8 Nn Nn 1c 1c ∼ | |
6 | simprr 733 | . . . . . . . . . 10 Nn Nn 1c 1c ∼ ∼ | |
7 | vex 2862 | . . . . . . . . . . 11 | |
8 | 7 | elcompl 3225 | . . . . . . . . . 10 ∼ |
9 | 6, 8 | sylib 188 | . . . . . . . . 9 Nn Nn 1c 1c ∼ |
10 | 7 | elsuci 4414 | . . . . . . . . . . . 12 1c |
11 | 8, 10 | sylan2b 461 | . . . . . . . . . . 11 ∼ 1c |
12 | 11 | adantl 452 | . . . . . . . . . 10 Nn Nn 1c 1c ∼ 1c |
13 | simplr 731 | . . . . . . . . . 10 Nn Nn 1c 1c ∼ 1c 1c | |
14 | 12, 13 | eleqtrd 2429 | . . . . . . . . 9 Nn Nn 1c 1c ∼ 1c |
15 | vex 2862 | . . . . . . . . . 10 | |
16 | 15, 7 | nnsucelr 4428 | . . . . . . . . 9 Nn 1c |
17 | 4, 9, 14, 16 | syl12anc 1180 | . . . . . . . 8 Nn Nn 1c 1c ∼ |
18 | nnceleq 4430 | . . . . . . . 8 Nn Nn | |
19 | 3, 4, 5, 17, 18 | syl22anc 1183 | . . . . . . 7 Nn Nn 1c 1c ∼ |
20 | 19 | a1d 22 | . . . . . 6 Nn Nn 1c 1c ∼ |
21 | 20 | rexlimdvva 2745 | . . . . 5 Nn Nn 1c 1c ∼ |
22 | 2, 21 | syl5bi 208 | . . . 4 Nn Nn 1c 1c 1c |
23 | 22 | exlimdv 1636 | . . 3 Nn Nn 1c 1c 1c |
24 | 1, 23 | syl5bi 208 | . 2 Nn Nn 1c 1c 1c |
25 | 24 | impr 602 | 1 Nn Nn 1c 1c 1c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 358 wex 1541 wceq 1642 wcel 1710 wne 2516 wrex 2615 ∼ ccompl 3205 cun 3207 c0 3550 csn 3737 1cc1c 4134 Nn cnnc 4373 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-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: preaddccan2 4455 evenodddisj 4516 peano4 4557 |
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