Theorem acexmidlemab 6043

Description: Lemma for acexmid 6048. (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 3511	. . . 4 ⊢ ¬ ∅ ∈ ∅
2		0ex 4236	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 3719	. . . . 5 ⊢ ∅ ∈ {∅}
4		eleq2 2296	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
5	3, 4	mpbiri 168	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
6	1, 5	mto 668	. . 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 6042	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = 𝐵)
11		id 19	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
12		eleq1 2295	. . . . . . . . . . . 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 2543	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
15	11, 14	riotaeqbidv 6005	. . . . . . . . 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 2241	. . . . . . 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 533	. . . . 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 2267	. . . 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 670	. 2 ⊢ (𝜑 → ¬ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))
24	23	con2i 632	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 716   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  {crab 2524  ∅c0 3507  {csn 3688  {cpr 3689  ℩crio 6001

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-ext 2214  ax-nul 4235

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-nul 3508  df-sn 3694  df-uni 3914  df-iota 5311  df-riota 6002

This theorem is referenced by:  acexmidlem1  6045