Theorem undifexmid 4285

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3574 and undifdcss 7120 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)

Hypothesis

Ref	Expression
undifexmid.1	⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)

Assertion

Ref	Expression
undifexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem undifexmid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4217	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 3701	. . . 4 ⊢ ∅ ∈ {∅}
3		ssrab2 3311	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
4		p0ex 4280	. . . . . . 7 ⊢ {∅} ∈ V
5	4	rabex 4235	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
6		sseq12 3251	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ⊆ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
7		simpl 109	. . . . . . . . 9 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑥 = {𝑧 ∈ {∅} ∣ 𝜑})
8		simpr 110	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑦 = {∅})
9	8, 7	difeq12d 3325	. . . . . . . . 9 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑦 ∖ 𝑥) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
10	7, 9	uneq12d 3361	. . . . . . . 8 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
11	10, 8	eqeq12d 2245	. . . . . . 7 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦 ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}))
12	6, 11	bibi12d 235	. . . . . 6 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})))
13		undifexmid.1	. . . . . 6 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)
14	5, 4, 12, 13	vtocl2 2858	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})
15	3, 14	mpbi 145	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}
16	2, 15	eleqtrri 2306	. . 3 ⊢ ∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
17		elun 3347	. . 3 ⊢ (∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) ↔ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
18	16, 17	mpbi 145	. 2 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
19		biidd 172	. . . . . 6 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
20	19	elrab3 2962	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
21	2, 20	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
22	21	biimpi 120	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
23		eldifn 3329	. . . 4 ⊢ (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
24	23, 21	sylnib 682	. . 3 ⊢ (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
25	22, 24	orim12i 766	. 2 ⊢ ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) → (𝜑 ∨ ¬ 𝜑))
26	18, 25	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2201  {crab 2513   ∖ cdif 3196   ∪ cun 3197   ⊆ wss 3199  ∅c0 3493  {csn 3670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676

This theorem is referenced by: (None)