Theorem exmidac 7467

Description: The axiom of choice implies excluded middle. See acexmid 6027 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 2238	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = ∅ ↔ 𝑥 = ∅))
2	1	orbi1d 799	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = ∅ ∨ 𝑦 = {∅}) ↔ (𝑥 = ∅ ∨ 𝑦 = {∅})))
3	2	cbvrabv 2802	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		eqeq1 2238	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = {∅} ↔ 𝑥 = {∅}))
5	4	orbi1d 799	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = {∅} ∨ 𝑦 = {∅}) ↔ (𝑥 = {∅} ∨ 𝑦 = {∅})))
6	5	cbvrabv 2802	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
7		eqid 2231	. 2 ⊢ {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}} = {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}}
8	3, 6, 7	exmidaclem 7466	1 ⊢ (CHOICE → EXMID)

Syntax hints:   → wi 4   ∨ wo 716   = wceq 1398  {crab 2515  ∅c0 3496  {csn 3673  {cpr 3674  EXMIDwem 4290  CHOICEwac 7463

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-exmid 4291  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ac 7464

This theorem is referenced by: (None)