Theorem exmidac 7186

Description: The axiom of choice implies excluded middle. See acexmid 5852 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 2177	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = ∅ ↔ 𝑥 = ∅))
2	1	orbi1d 786	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = ∅ ∨ 𝑦 = {∅}) ↔ (𝑥 = ∅ ∨ 𝑦 = {∅})))
3	2	cbvrabv 2729	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		eqeq1 2177	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 = {∅} ↔ 𝑥 = {∅}))
5	4	orbi1d 786	. . 3 ⊢ (𝑢 = 𝑥 → ((𝑢 = {∅} ∨ 𝑦 = {∅}) ↔ (𝑥 = {∅} ∨ 𝑦 = {∅})))
6	5	cbvrabv 2729	. 2 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
7		eqid 2170	. 2 ⊢ {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}} = {{𝑢 ∈ {∅, {∅}} ∣ (𝑢 = ∅ ∨ 𝑦 = {∅})}, {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ 𝑦 = {∅})}}
8	3, 6, 7	exmidaclem 7185	1 ⊢ (CHOICE → EXMID)

Syntax hints:   → wi 4   ∨ wo 703   = wceq 1348  {crab 2452  ∅c0 3414  {csn 3583  {cpr 3584  EXMIDwem 4180  CHOICEwac 7182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-exmid 4181  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ac 7183

This theorem is referenced by: (None)