Theorem exmidac 7337

Description: The axiom of choice implies excluded middle. See acexmid 5956 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 2213	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = ∅ ↔ 𝑥 = ∅))
2	1	orbi1d 793	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = ∅ ∨ 𝑦 = {∅}) ↔ (𝑥 = ∅ ∨ 𝑦 = {∅})))
3	2	cbvrabv 2772	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		eqeq1 2213	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = {∅} ↔ 𝑥 = {∅}))
5	4	orbi1d 793	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = {∅} ∨ 𝑦 = {∅}) ↔ (𝑥 = {∅} ∨ 𝑦 = {∅})))
6	5	cbvrabv 2772	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
7		eqid 2206	. 2 ⊢ {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}} = {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}}
8	3, 6, 7	exmidaclem 7336	1 ⊢ (CHOICE → EXMID)

Syntax hints:   → wi 4   ∨ wo 710   = wceq 1373  {crab 2489  ∅c0 3464  {csn 3638  {cpr 3639  EXMIDwem 4246  CHOICEwac 7333

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-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-exmid 4247  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ac 7334

This theorem is referenced by: (None)