MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqsnOLD Structured version   Visualization version   GIF version

Theorem eqsnOLD 4330
Description: Obsolete proof of eqsn 4329 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 3636 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
2 df-ne 2791 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3 sssn 4326 . . . . . . 7 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
43biimpi 206 . . . . . 6 (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
54ord 392 . . . . 5 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
62, 5syl5bi 232 . . . 4 (𝐴 ⊆ {𝐵} → (𝐴 ≠ ∅ → 𝐴 = {𝐵}))
76com12 32 . . 3 (𝐴 ≠ ∅ → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵}))
81, 7impbid2 216 . 2 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ 𝐴 ⊆ {𝐵}))
9 dfss3 3573 . . 3 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝐵})
10 velsn 4164 . . . 4 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
1110ralbii 2974 . . 3 (∀𝑥𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
129, 11bitri 264 . 2 (𝐴 ⊆ {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵)
138, 12syl6bb 276 1 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383   = wceq 1480  wcel 1987  wne 2790  wral 2907  wss 3555  c0 3891  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator