Theorem undifexmid 4190

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3503 and undifdcss 6916 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 4127	. . . . 5 ⊢ ∅ ∈ V
2	1	snid 3622	. . . 4 ⊢ ∅ ∈ {∅}
3		ssrab2 3240	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
4		p0ex 4185	. . . . . . 7 ⊢ {∅} ∈ V
5	4	rabex 4144	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
6		sseq12 3180	. . . . . . 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 3254	. . . . . . . . 9 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑦 ∖ 𝑥) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
10	7, 9	uneq12d 3290	. . . . . . . 8 ⊢ ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦 ∖ 𝑥)) = ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
11	10, 8	eqeq12d 2192	. . . . . . 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 2792	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})
15	3, 14	mpbi 145	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}
16	2, 15	eleqtrri 2253	. . 3 ⊢ ∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
17		elun 3276	. . 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 2894	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
21	2, 20	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
22	21	biimpi 120	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
23		eldifn 3258	. . . 4 ⊢ (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
24	23, 21	sylnib 676	. . 3 ⊢ (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
25	22, 24	orim12i 759	. 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 708   = wceq 1353   ∈ wcel 2148  {crab 2459   ∖ cdif 3126   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  {csn 3591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597

This theorem is referenced by: (None)