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Mirrors > Home > MPE Home > Th. List > ax1ne0 | Structured version Visualization version GIF version |
Description: 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 10609. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax1ne0 | ⊢ 1 ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ne0sr 10521 | . . . 4 ⊢ ¬ 1R = 0R | |
2 | 1sr 10506 | . . . . . 6 ⊢ 1R ∈ R | |
3 | 2 | elexi 3516 | . . . . 5 ⊢ 1R ∈ V |
4 | 3 | eqresr 10562 | . . . 4 ⊢ (〈1R, 0R〉 = 〈0R, 0R〉 ↔ 1R = 0R) |
5 | 1, 4 | mtbir 325 | . . 3 ⊢ ¬ 〈1R, 0R〉 = 〈0R, 0R〉 |
6 | df-1 10548 | . . . 4 ⊢ 1 = 〈1R, 0R〉 | |
7 | df-0 10547 | . . . 4 ⊢ 0 = 〈0R, 0R〉 | |
8 | 6, 7 | eqeq12i 2839 | . . 3 ⊢ (1 = 0 ↔ 〈1R, 0R〉 = 〈0R, 0R〉) |
9 | 5, 8 | mtbir 325 | . 2 ⊢ ¬ 1 = 0 |
10 | 9 | neir 3022 | 1 ⊢ 1 ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ≠ wne 3019 〈cop 4576 Rcnr 10290 0Rc0r 10291 1Rc1r 10292 0cc0 10540 1c1 10541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-ec 8294 df-qs 8298 df-ni 10297 df-pli 10298 df-mi 10299 df-lti 10300 df-plpq 10333 df-mpq 10334 df-ltpq 10335 df-enq 10336 df-nq 10337 df-erq 10338 df-plq 10339 df-mq 10340 df-1nq 10341 df-rq 10342 df-ltnq 10343 df-np 10406 df-1p 10407 df-plp 10408 df-ltp 10410 df-enr 10480 df-nr 10481 df-ltr 10484 df-0r 10485 df-1r 10486 df-0 10547 df-1 10548 |
This theorem is referenced by: (None) |
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