Theorem en1eqsnbi 9180

Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20748. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)

Assertion

Ref	Expression
en1eqsnbi	⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴}))

Proof of Theorem en1eqsnbi

Step	Hyp	Ref	Expression
1		en1eqsn 9179	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})
2	1	ex 412	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = {𝐴}))
3		ensn1g 8963	. . 3 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o)
4		breq1 5089	. . 3 ⊢ (𝐵 = {𝐴} → (𝐵 ≈ 1o ↔ {𝐴} ≈ 1o))
5	3, 4	syl5ibrcom 247	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = {𝐴} → 𝐵 ≈ 1o))
6	2, 5	impbid 212	1 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴}))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {csn 4568   class class class wbr 5086  1_oc1o 8392   ≈ cen 8884

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-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

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-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1o 8399  df-en 8888

This theorem is referenced by:  srgen1zr  20191  rngen1zr  20748  rngosn4  38263