Theorem modom 6993

Description: Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
modom	⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o)

Proof of Theorem modom

Dummy variables 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 6589	. . 3 ⊢ 1o ∈ V
2		0lt1o 6607	. . . . 5 ⊢ ∅ ∈ 1o
3	2	2a1i 27	. . . 4 ⊢ (∃*𝑥𝜑 → (𝑢 ∈ {𝑥 ∣ 𝜑} → ∅ ∈ 1o))
4		eqidd 2232	. . . . . 6 ⊢ ((∃*𝑥𝜑 ∧ (𝑢 ∈ {𝑥 ∣ 𝜑} ∧ 𝑣 ∈ {𝑥 ∣ 𝜑})) → ∅ = ∅)
5		df-clab 2218	. . . . . . . . 9 ⊢ (𝑢 ∈ {𝑥 ∣ 𝜑} ↔ [𝑢 / 𝑥]𝜑)
6		df-clab 2218	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑥 ∣ 𝜑} ↔ [𝑣 / 𝑥]𝜑)
7	5, 6	anbi12i 460	. . . . . . . 8 ⊢ ((𝑢 ∈ {𝑥 ∣ 𝜑} ∧ 𝑣 ∈ {𝑥 ∣ 𝜑}) ↔ ([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑))
8		nfv 1576	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜑
9	8	mo3 2134	. . . . . . . . . 10 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
10		nfcv 2374	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑢
11		nfs1v 1992	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥[𝑢 / 𝑥]𝜑
12		nfs1v 1992	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
13	11, 12	nfan 1613	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)
14		nfv 1576	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝑢 = 𝑦
15	13, 14	nfim 1620	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦)
16	15	nfal 1624	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦)
17		sbequ12 1819	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝜑 ↔ [𝑢 / 𝑥]𝜑))
18	17	anbi1d 465	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑)))
19		equequ1 1760	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
20	18, 19	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦)))
21	20	albidv 1872	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦)))
22	10, 16, 21	spcgf 2888	. . . . . . . . . . 11 ⊢ (𝑢 ∈ V → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦)))
23	22	elv 2806	. . . . . . . . . 10 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦))
24	9, 23	sylbi 121	. . . . . . . . 9 ⊢ (∃*𝑥𝜑 → ∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦))
25		nfcv 2374	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑣
26		nfv 1576	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑) → 𝑢 = 𝑣)
27		sbequ 1888	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → ([𝑦 / 𝑥]𝜑 ↔ [𝑣 / 𝑥]𝜑))
28	27	anbi2d 464	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑)))
29		equequ2 1761	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑢 = 𝑦 ↔ 𝑢 = 𝑣))
30	28, 29	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → ((([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦) ↔ (([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑) → 𝑢 = 𝑣)))
31	25, 26, 30	spcgf 2888	. . . . . . . . . 10 ⊢ (𝑣 ∈ V → (∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦) → (([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑) → 𝑢 = 𝑣)))
32	31	elv 2806	. . . . . . . . 9 ⊢ (∀𝑦(([𝑢 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑢 = 𝑦) → (([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑) → 𝑢 = 𝑣))
33	24, 32	syl 14	. . . . . . . 8 ⊢ (∃*𝑥𝜑 → (([𝑢 / 𝑥]𝜑 ∧ [𝑣 / 𝑥]𝜑) → 𝑢 = 𝑣))
34	7, 33	biimtrid 152	. . . . . . 7 ⊢ (∃*𝑥𝜑 → ((𝑢 ∈ {𝑥 ∣ 𝜑} ∧ 𝑣 ∈ {𝑥 ∣ 𝜑}) → 𝑢 = 𝑣))
35	34	imp 124	. . . . . 6 ⊢ ((∃*𝑥𝜑 ∧ (𝑢 ∈ {𝑥 ∣ 𝜑} ∧ 𝑣 ∈ {𝑥 ∣ 𝜑})) → 𝑢 = 𝑣)
36	4, 35	2thd 175	. . . . 5 ⊢ ((∃*𝑥𝜑 ∧ (𝑢 ∈ {𝑥 ∣ 𝜑} ∧ 𝑣 ∈ {𝑥 ∣ 𝜑})) → (∅ = ∅ ↔ 𝑢 = 𝑣))
37	36	ex 115	. . . 4 ⊢ (∃*𝑥𝜑 → ((𝑢 ∈ {𝑥 ∣ 𝜑} ∧ 𝑣 ∈ {𝑥 ∣ 𝜑}) → (∅ = ∅ ↔ 𝑢 = 𝑣)))
38	3, 37	dom2d 6945	. . 3 ⊢ (∃*𝑥𝜑 → (1o ∈ V → {𝑥 ∣ 𝜑} ≼ 1o))
39	1, 38	mpi 15	. 2 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ≼ 1o)
40		nfab1 2376	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
41		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑥 ≼
42		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑥1o
43	40, 41, 42	nfbr 4135	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} ≼ 1o
44		simpl 109	. . . . . . 7 ⊢ (({𝑥 ∣ 𝜑} ≼ 1o ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → {𝑥 ∣ 𝜑} ≼ 1o)
45		abid 2219	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
46	45	biimpri 133	. . . . . . . 8 ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∣ 𝜑})
47	46	ad2antrl 490	. . . . . . 7 ⊢ (({𝑥 ∣ 𝜑} ≼ 1o ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 ∈ {𝑥 ∣ 𝜑})
48		df-clab 2218	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
49	48	biimpri 133	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})
50	49	ad2antll 491	. . . . . . 7 ⊢ (({𝑥 ∣ 𝜑} ≼ 1o ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑦 ∈ {𝑥 ∣ 𝜑})
51		1dom1el 6992	. . . . . . 7 ⊢ (({𝑥 ∣ 𝜑} ≼ 1o ∧ 𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) → 𝑥 = 𝑦)
52	44, 47, 50, 51	syl3anc 1273	. . . . . 6 ⊢ (({𝑥 ∣ 𝜑} ≼ 1o ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦)
53	52	ex 115	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} ≼ 1o → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
54	53	alrimiv 1922	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ≼ 1o → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
55	43, 54	alrimi 1570	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≼ 1o → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
56	55, 9	sylibr 134	. 2 ⊢ ({𝑥 ∣ 𝜑} ≼ 1o → ∃*𝑥𝜑)
57	39, 56	impbii 126	1 ⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   = wceq 1397  [wsb 1810  ∃*wmo 2080   ∈ wcel 2202  {cab 2217  Vcvv 2802  ∅c0 3494   class class class wbr 4088  1_oc1o 6574   ≼ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-dom 6910

This theorem is referenced by:  modom2  6994