Theorem acexmidlemab 5894

Description: Lemma for acexmid 5899. (Contributed by Jim Kingdon, 6-Aug-2019.)

Hypotheses

Ref	Expression
acexmidlem.a	⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
acexmidlem.b	⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
acexmidlem.c	⊢ 𝐶 = {𝐴, 𝐵}

Assertion

Ref	Expression
acexmidlemab	⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ¬ 𝜑)

Distinct variable groups:   𝑥,𝑦,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑣,𝑢   𝑥,𝐶,𝑦,𝑣,𝑢   𝜑,𝑥,𝑦,𝑣,𝑢

Proof of Theorem acexmidlemab

Step	Hyp	Ref	Expression
1		noel 3441	. . . 4 ⊢ ¬ ∅ ∈ ∅
2		0ex 4148	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 3641	. . . . 5 ⊢ ∅ ∈ {∅}
4		eleq2 2253	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
5	3, 4	mpbiri 168	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
6	1, 5	mto 663	. . 3 ⊢ ¬ ∅ = {∅}
7		acexmidlem.a	. . . . . . . . . 10 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
8		acexmidlem.b	. . . . . . . . . 10 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
9		acexmidlem.c	. . . . . . . . . 10 ⊢ 𝐶 = {𝐴, 𝐵}
10	7, 8, 9	acexmidlemph 5893	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = 𝐵)
11		id 19	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
12		eleq1 2252	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
13	12	anbi1d 465	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
14	13	rexbidv 2491	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
15	11, 14	riotaeqbidv 5858	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
16	10, 15	syl 14	. . . . . . . 8 ⊢ (𝜑 → (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
17	16	eqeq1d 2198	. . . . . . 7 ⊢ (𝜑 → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ↔ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅))
18	17	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅) → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅)
19	18	adantrr 479	. . . . 5 ⊢ ((𝜑 ∧ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅)
20		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})
21	19, 20	eqtr3d 2224	. . . 4 ⊢ ((𝜑 ∧ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → ∅ = {∅})
22	21	ex 115	. . 3 ⊢ (𝜑 → (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ∅ = {∅}))
23	6, 22	mtoi 665	. 2 ⊢ (𝜑 → ¬ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
24	23	con2i 628	1 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2160  ∃wrex 2469  {crab 2472  ∅c0 3437  {csn 3610  {cpr 3611  ℩crio 5854

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4147

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-nul 3438  df-sn 3616  df-uni 3828  df-iota 5199  df-riota 5855

This theorem is referenced by:  acexmidlem1  5896