Theorem exmid1dc 4123

Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4117 or ordtriexmid 4437. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)

Hypothesis

Ref	Expression
exmid1dc.x	⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})

Assertion

Ref	Expression
exmid1dc	⊢ (𝜑 → EXMID)

Distinct variable group:   𝜑,𝑥

Proof of Theorem exmid1dc

Step	Hyp	Ref	Expression
1		exmid1dc.x	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})
2		exmiddc 821	. . . . . . 7 ⊢ (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
3	1, 2	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4		df-ne 2309	. . . . . . . . 9 ⊢ (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5		pwntru 4122	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
6	5	ex 114	. . . . . . . . 9 ⊢ (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
7	4, 6	syl5bir 152	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
8	7	orim2d 777	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
9	8	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
10	3, 9	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
11	10	orcomd 718	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
12	11	ex 114	. . 3 ⊢ (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
13	12	alrimiv 1846	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		exmid01 4121	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
15	13, 14	sylibr 133	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819  ∀wal 1329   = wceq 1331   ≠ wne 2308   ⊆ wss 3071  ∅c0 3363  {csn 3527  EXMIDwem 4118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-exmid 4119

This theorem is referenced by:  exmidonfin  7050  exmidaclem  7064