Theorem 1div0OLD 11777

Description: Obsolete version of 1div0 11776 as of 5-Jun-2025. (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
1div0OLD	⊢ (1 / 0) = ∅

Proof of Theorem 1div0OLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-div 11775	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		riotaex 7307	. . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
3	1, 2	dmmpo 8003	. 2 ⊢ dom / = (ℂ × (ℂ ∖ {0}))
4		eqid 2731	. . 3 ⊢ 0 = 0
5		eldifsni 4742	. . . . 5 ⊢ (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0)
6	5	adantl 481	. . . 4 ⊢ ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0)
7	6	necon2bi 2958	. . 3 ⊢ (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})))
8	4, 7	ax-mp 5	. 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
9		ndmovg 7529	. 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
10	3, 8, 9	mp2an 692	1 ⊢ (1 / 0) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   ∖ cdif 3899  ∅c0 4283  {csn 4576   × cxp 5614  dom cdm 5616  ℩crio 7302  (class class class)co 7346  ℂcc 11004  0cc0 11006  1c1 11007   · cmul 11011   / cdiv 11774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-div 11775

This theorem is referenced by: (None)