Theorem en1bg 6969

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 6968	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 19	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 3900	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 2803	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 3907	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	eqtrdi 2278	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 3680	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2265	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1644	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
10	1, 9	sylbi 121	. 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴})
11		uniexg 4534	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
12		ensn1g 6966	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o)
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ∈ 𝑉 → {∪ 𝐴} ≈ 1o)
14		breq1 4089	. . 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 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  {csn 3667  ∪ cuni 3891   class class class wbr 4086  1_oc1o 6570   ≈ cen 6902

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-en 6905

This theorem is referenced by:  en1uniel  6973