Theorem exmid1dc 4249

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 4242 or ordtriexmid 4574. 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 838	. . . . . . 7 ⊢ (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
3	1, 2	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4		df-ne 2378	. . . . . . . . 9 ⊢ (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5		pwntru 4248	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
6	5	ex 115	. . . . . . . . 9 ⊢ (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
7	4, 6	biimtrrid 153	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
8	7	orim2d 790	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
9	8	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
10	3, 9	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
11	10	orcomd 731	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
12	11	ex 115	. . 3 ⊢ (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
13	12	alrimiv 1898	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		exmid01 4247	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
15	13, 14	sylibr 134	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  DECID wdc 836  ∀wal 1371   = wceq 1373   ≠ wne 2377   ⊆ wss 3168  ∅c0 3462  {csn 3635  EXMIDwem 4243

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-nul 4175

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-v 2775  df-dif 3170  df-in 3174  df-ss 3181  df-nul 3463  df-sn 3641  df-exmid 4244

This theorem is referenced by:  pw1fin  7019  exmidonfin  7315  exmidaclem  7333  exmidontri  7364  exmidontri2or  7368