Theorem iunopeqop 5464

Description: Implication of an ordered pair being equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.) Remove antecedent. (Revised by Eric Schmidt, 9-May-2026.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
iunopeqop.b	⊢ 𝐵 ∈ V
iunopeqop.c	⊢ 𝐶 ∈ V
iunopeqop.d	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
iunopeqop	⊢ (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐵(𝑥,𝑧)   𝐶(𝑧)   𝐷(𝑧)

Proof of Theorem iunopeqop

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunopeqop.c	. . . . . . 7 ⊢ 𝐶 ∈ V
2		iunopeqop.d	. . . . . . 7 ⊢ 𝐷 ∈ V
3	1, 2	opnzi 5416	. . . . . 6 ⊢ ⟨𝐶, 𝐷⟩ ≠ ∅
4	3	a1i 11	. . . . 5 ⊢ (𝐴 = ∅ → ⟨𝐶, 𝐷⟩ ≠ ∅)
5		iuneq1 4940	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ∪ 𝑥 ∈ ∅ {⟨𝑥, 𝐵⟩})
6		0iun 4994	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ {⟨𝑥, 𝐵⟩} = ∅
7	5, 6	eqtrdi 2786	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ∅)
8	4, 7	neeqtrrd 3004	. . . 4 ⊢ (𝐴 = ∅ → ⟨𝐶, 𝐷⟩ ≠ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩})
9		nesym 2986	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ≠ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} ↔ ¬ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩)
10	8, 9	sylib 218	. . 3 ⊢ (𝐴 = ∅ → ¬ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩)
11	10	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
12		n0snor2el 4766	. . 3 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
13		nfiu1 4959	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
14	13	nfeq1 2912	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩
15		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥∃𝑧 𝐴 = {𝑧}
16	14, 15	nfim 1898	. . . . 5 ⊢ Ⅎ𝑥(∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})
17		ssiun2 4979	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → {⟨𝑥, 𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩})
18		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
19		nfcsb1v 3857	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
20	18, 19	nfop 4822	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩
21	20	nfsn 4641	. . . . . . . . . 10 ⊢ Ⅎ𝑥{⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}
22	21, 13	nfss 3910	. . . . . . . . 9 ⊢ Ⅎ𝑥{⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
23		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
24		csbeq1a 3847	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
25	23, 24	opeq12d 4814	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩)
26	25	sneqd 4569	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → {⟨𝑥, 𝐵⟩} = {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩})
27	26	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ({⟨𝑥, 𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} ↔ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}))
28	18, 22, 27, 17	vtoclgaf 3517	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩})
29	17, 28	anim12i 614	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({⟨𝑥, 𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}))
30		unss 4121	. . . . . . . 8 ⊢ (({⟨𝑥, 𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}) ↔ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩})
31		sseq2 3943	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} ↔ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩))
32		df-pr 4560	. . . . . . . . . . . . 13 ⊢ {⟨𝑥, 𝐵⟩, ⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} = ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩})
33	32	eqcomi 2744	. . . . . . . . . . . 12 ⊢ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) = {⟨𝑥, 𝐵⟩, ⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}
34	33	sseq1i 3945	. . . . . . . . . . 11 ⊢ (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩ ↔ {⟨𝑥, 𝐵⟩, ⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩)
35		vex 3431	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
36		iunopeqop.b	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ V
37		vex 3431	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
38	36	csbex 5235	. . . . . . . . . . . . 13 ⊢ ⦋𝑦 / 𝑥⦌𝐵 ∈ V
39	35, 36, 37, 38, 1, 2	propssopi 5451	. . . . . . . . . . . 12 ⊢ ({⟨𝑥, 𝐵⟩, ⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩ → 𝑥 = 𝑦)
40		eqneqall 2941	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ≠ 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 𝐴 = {𝑧})))
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ ({⟨𝑥, 𝐵⟩, ⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩ → (𝑥 ≠ 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 𝐴 = {𝑧})))
42	34, 41	sylbi 217	. . . . . . . . . 10 ⊢ (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩ → (𝑥 ≠ 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 𝐴 = {𝑧})))
43	31, 42	biimtrdi 253	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} → (𝑥 ≠ 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 𝐴 = {𝑧}))))
44	43	com14 96	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩}) ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} → (𝑥 ≠ 𝑦 → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))))
45	30, 44	biimtrid 242	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (({⟨𝑥, 𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, ⦋𝑦 / 𝑥⦌𝐵⟩} ⊆ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}) → (𝑥 ≠ 𝑦 → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))))
46	29, 45	mpd 15	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})))
47	46	rexlimdva 3136	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})))
48	16, 47	rexlimi 3235	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
49		ax-1 6	. . . 4 ⊢ (∃𝑧 𝐴 = {𝑧} → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
50	48, 49	jaoi 858	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}) → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
51	12, 50	syl 17	. 2 ⊢ (𝐴 ≠ ∅ → (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
52	11, 51	pm2.61ine 3013	1 ⊢ (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2930  ∃wrex 3059  Vcvv 3427  ⦋csb 3833   ∪ cun 3883   ⊆ wss 3885  ∅c0 4263  {csn 4557  {cpr 4559  ⟨cop 4563  ∪ ciun 4923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4925

This theorem is referenced by:  funopsn  7090