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Mirrors > Home > NFE Home > Th. List > 0cnsuc | GIF version |
Description: Cardinal zero is not a successor. Compare Theorem X.1.2 of [Rosser] p. 275. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
0cnsuc | ⊢ (A +c 1c) ≠ 0c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelsuc 4400 | . . 3 ⊢ ¬ ∅ ∈ (A +c 1c) | |
2 | 0ex 4110 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | snid 3760 | . . . . 5 ⊢ ∅ ∈ {∅} |
4 | df-0c 4377 | . . . . 5 ⊢ 0c = {∅} | |
5 | 3, 4 | eleqtrri 2426 | . . . 4 ⊢ ∅ ∈ 0c |
6 | eleq2 2414 | . . . 4 ⊢ ((A +c 1c) = 0c → (∅ ∈ (A +c 1c) ↔ ∅ ∈ 0c)) | |
7 | 5, 6 | mpbiri 224 | . . 3 ⊢ ((A +c 1c) = 0c → ∅ ∈ (A +c 1c)) |
8 | 1, 7 | mto 167 | . 2 ⊢ ¬ (A +c 1c) = 0c |
9 | df-ne 2518 | . 2 ⊢ ((A +c 1c) ≠ 0c ↔ ¬ (A +c 1c) = 0c) | |
10 | 8, 9 | mpbir 200 | 1 ⊢ (A +c 1c) ≠ 0c |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 {csn 3737 1cc1c 4134 0cc0c 4374 +c 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-0c 4377 df-addc 4378 |
This theorem is referenced by: peano3 4404 ltfinirr 4457 evenodddisj 4516 0cnelphi 4597 addceq0 6219 1ne0c 6241 2ne0c 6242 nnltp1c 6262 nnc3n3p1 6278 nchoicelem12 6300 nchoicelem14 6302 nchoicelem17 6305 fnfreclem2 6318 fnfreclem3 6319 |
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