Theorem en1bg 6216

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)

Assertion

Ref	Expression
en1bg	⊢ (A ∈ 𝑉 → (A ≈ 1𝑜 ↔ A = {∪ A}))

Proof of Theorem en1bg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 6215	. . 3 ⊢ (A ≈ 1𝑜 ↔ ∃x A = {x})
2		id 19	. . . . 5 ⊢ (A = {x} → A = {x})
3		unieq 3580	. . . . . . 7 ⊢ (A = {x} → ∪ A = ∪ {x})
4		vex 2554	. . . . . . . 8 ⊢ x ∈ V
5	4	unisn 3587	. . . . . . 7 ⊢ ∪ {x} = x
6	3, 5	syl6eq 2085	. . . . . 6 ⊢ (A = {x} → ∪ A = x)
7	6	sneqd 3380	. . . . 5 ⊢ (A = {x} → {∪ A} = {x})
8	2, 7	eqtr4d 2072	. . . 4 ⊢ (A = {x} → A = {∪ A})
9	8	exlimiv 1486	. . 3 ⊢ (∃x A = {x} → A = {∪ A})
10	1, 9	sylbi 114	. 2 ⊢ (A ≈ 1𝑜 → A = {∪ A})
11		uniexg 4141	. . . 4 ⊢ (A ∈ 𝑉 → ∪ A ∈ V)
12		ensn1g 6213	. . . 4 ⊢ (∪ A ∈ V → {∪ A} ≈ 1𝑜)
13	11, 12	syl 14	. . 3 ⊢ (A ∈ 𝑉 → {∪ A} ≈ 1𝑜)
14		breq1 3758	. . 3 ⊢ (A = {∪ A} → (A ≈ 1𝑜 ↔ {∪ A} ≈ 1𝑜))
15	13, 14	syl5ibrcom 146	. 2 ⊢ (A ∈ 𝑉 → (A = {∪ A} → A ≈ 1𝑜))
16	10, 15	impbid2 131	1 ⊢ (A ∈ 𝑉 → (A ≈ 1𝑜 ↔ A = {∪ A}))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  {csn 3367  ∪ cuni 3571   class class class wbr 3755  1_𝑜c1o 5933   ≈ cen 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1o 5940  df-en 6158

This theorem is referenced by:  en1uniel  6220