Theorem acexmidlemab 5776

Description: Lemma for acexmid 5781. (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 3372	. . . 4 ⊢ ¬ ∅ ∈ ∅
2		0ex 4063	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 3563	. . . . 5 ⊢ ∅ ∈ {∅}
4		eleq2 2204	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
5	3, 4	mpbiri 167	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
6	1, 5	mto 652	. . 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 5775	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = 𝐵)
11		id 19	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
12		eleq1 2203	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
13	12	anbi1d 461	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
14	13	rexbidv 2439	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
15	11, 14	riotaeqbidv 5741	. . . . . . . . 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 2149	. . . . . . 7 ⊢ (𝜑 → ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ↔ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅))
18	17	biimpa 294	. . . . . 6 ⊢ ((𝜑 ∧ (℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅) → (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅)
19	18	adantrr 471	. . . . 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 522	. . . . 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 2175	. . . 4 ⊢ ((𝜑 ∧ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → ∅ = {∅})
22	21	ex 114	. . 3 ⊢ (𝜑 → (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ∅ = {∅}))
23	6, 22	mtoi 654	. 2 ⊢ (𝜑 → ¬ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
24	23	con2i 617	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 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  {crab 2421  ∅c0 3368  {csn 3532  {cpr 3533  ℩crio 5737

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-nul 3369  df-sn 3538  df-uni 3745  df-iota 5096  df-riota 5738

This theorem is referenced by:  acexmidlem1  5778