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Theorem prssOLD 4384
Description: Obsolete proof of prss 4383 as of 23-Jul-2021. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
prss.1 𝐴 ∈ V
prss.2 𝐵 ∈ V
Assertion
Ref Expression
prssOLD ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Proof of Theorem prssOLD
StepHypRef Expression
1 unss 3820 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2 prss.1 . . . 4 𝐴 ∈ V
32snss 4348 . . 3 (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶)
4 prss.2 . . . 4 𝐵 ∈ V
54snss 4348 . . 3 (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶)
63, 5anbi12i 733 . 2 ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7 df-pr 4213 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
87sseq1i 3662 . 2 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
91, 6, 83bitr4i 292 1 ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  Vcvv 3231  cun 3605  wss 3607  {csn 4210  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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