Theorem en1bg 7054

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 7053	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 19	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 3929	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 3936	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	eqtrdi 2283	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 3708	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2270	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1647	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
10	1, 9	sylbi 121	. 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴})
11		uniexg 4566	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
12		ensn1g 7051	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o)
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ∈ 𝑉 → {∪ 𝐴} ≈ 1o)
14		breq1 4118	. . 3 ⊢ (𝐴 = {∪ 𝐴} → (𝐴 ≈ 1o ↔ {∪ 𝐴} ≈ 1o))
15	13, 14	syl5ibrcom 157	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o))
16	10, 15	impbid2 143	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  {csn 3695  ∪ cuni 3920   class class class wbr 4115  1_oc1o 6654   ≈ cen 6987

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-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-id 4420  df-suc 4498  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1o 6661  df-en 6990

This theorem is referenced by:  en1uniel  7058