Theorem 2p2ne5 49382

Description: Prove that 2 + 2 ≠ 5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase 2 + 2 = 5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)

Assertion

Ref	Expression
2p2ne5	⊢ (2 + 2) ≠ 5

Proof of Theorem 2p2ne5

Step	Hyp	Ref	Expression
1		2p2e4 12367	. 2 ⊢ (2 + 2) = 4
2		4re 12316	. . 3 ⊢ 4 ∈ ℝ
3		4lt5 12409	. . 3 ⊢ 4 < 5
4	2, 3	ltneii 11340	. 2 ⊢ 4 ≠ 5
5	1, 4	eqnetri 3001	1 ⊢ (2 + 2) ≠ 5

Syntax hints:   ≠ wne 2931  (class class class)co 7399   + caddc 11124  2c2 12287  4c4 12289  5c5 12290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-resscn 11178  ax-1cn 11179  ax-icn 11180  ax-addcl 11181  ax-addrcl 11182  ax-mulcl 11183  ax-mulrcl 11184  ax-mulcom 11185  ax-addass 11186  ax-mulass 11187  ax-distr 11188  ax-i2m1 11189  ax-1ne0 11190  ax-1rid 11191  ax-rnegex 11192  ax-rrecex 11193  ax-cnre 11194  ax-pre-lttri 11195  ax-pre-lttrn 11196  ax-pre-ltadd 11197  ax-pre-mulgt0 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-po 5558  df-so 5559  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-er 8713  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11460  df-neg 11461  df-2 12295  df-3 12296  df-4 12297  df-5 12298

This theorem is referenced by: (None)