Theorem exmidac 7276

Description: The axiom of choice implies excluded middle. See acexmid 5921 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 2203	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = ∅ ↔ 𝑥 = ∅))
2	1	orbi1d 792	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = ∅ ∨ 𝑦 = {∅}) ↔ (𝑥 = ∅ ∨ 𝑦 = {∅})))
3	2	cbvrabv 2762	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		eqeq1 2203	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = {∅} ↔ 𝑥 = {∅}))
5	4	orbi1d 792	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = {∅} ∨ 𝑦 = {∅}) ↔ (𝑥 = {∅} ∨ 𝑦 = {∅})))
6	5	cbvrabv 2762	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
7		eqid 2196	. 2 ⊢ {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}} = {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}}
8	3, 6, 7	exmidaclem 7275	1 ⊢ (CHOICE → EXMID)

Syntax hints:   → wi 4   ∨ wo 709   = wceq 1364  {crab 2479  ∅c0 3450  {csn 3622  {cpr 3623  EXMIDwem 4227  CHOICEwac 7272

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-exmid 4228  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ac 7273

This theorem is referenced by: (None)