Theorem exmidac 7571

Description: The axiom of choice implies excluded middle. See acexmid 6087 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)

Assertion

Ref	Expression
exmidac	⊢ (CHOICE → EXMID)

Proof of Theorem exmidac

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

Step	Hyp	Ref	Expression
1		eqeq1 2245	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = ∅ ↔ 𝑥 = ∅))
2	1	orbi1d 803	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = ∅ ∨ 𝑦 = {∅}) ↔ (𝑥 = ∅ ∨ 𝑦 = {∅})))
3	2	cbvrabv 2820	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		eqeq1 2245	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = {∅} ↔ 𝑥 = {∅}))
5	4	orbi1d 803	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = {∅} ∨ 𝑦 = {∅}) ↔ (𝑥 = {∅} ∨ 𝑦 = {∅})))
6	5	cbvrabv 2820	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
7		eqid 2238	. 2 ⊢ {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}} = {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}}
8	3, 6, 7	exmidaclem 7570	1 ⊢ (CHOICE → EXMID)

This proof depends on syntax axioms:   → wi 4   ∨ wo 720   = wceq 1402  {crab 2532  ∅c0 3520  {csn 3709  {cpr 3710  EXMIDwem 4333  CHOICEwac 7567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-exmid 4334  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ac 7568

This theorem is used by: (None)