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Theorem eqsnOLD 4302
 Description: Obsolete proof of eqsn 4301 as of 23-Jul-2021. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
eqsnOLD (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqsnOLD
StepHypRef Expression
1 eqimss 3620 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
2 df-ne 2782 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 sssn 4298 . . . . . . 7 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
43biimpi 205 . . . . . 6 (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
54ord 391 . . . . 5 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
62, 5syl5bi 231 . . . 4 (𝐴 ⊆ {𝐵} → (𝐴 ≠ ∅ → 𝐴 = {𝐵}))
76com12 32 . . 3 (𝐴 ≠ ∅ → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵}))
81, 7impbid2 215 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ 𝐴 ⊆ {𝐵}))
9 dfss3 3558 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
10 velsn 4141 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
1110ralbii 2963 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
129, 11bitri 263 . 2 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
138, 12syl6bb 275 1 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126 This theorem is referenced by: (None)
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