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Theorem zfpair 4831
 Description: The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms. This theorem should not be referenced by any proof other than axpr 4832. Instead, use zfpair2 4834 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
Assertion
Ref Expression
zfpair {𝑥, 𝑦} ∈ V

Proof of Theorem zfpair
Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfpr2 4143 . 2 {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)}
2 19.43 1799 . . . . 5 (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)))
3 prlem2 998 . . . . . 6 (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
43exbii 1764 . . . . 5 (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5 0ex 4718 . . . . . . . 8 ∅ ∈ V
65isseti 3182 . . . . . . 7 𝑧 𝑧 = ∅
7 19.41v 1901 . . . . . . 7 (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
86, 7mpbiran 955 . . . . . 6 (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9 p0ex 4779 . . . . . . . 8 {∅} ∈ V
109isseti 3182 . . . . . . 7 𝑧 𝑧 = {∅}
11 19.41v 1901 . . . . . . 7 (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ (∃𝑧 𝑧 = {∅} ∧ 𝑤 = 𝑦))
1210, 11mpbiran 955 . . . . . 6 (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)
138, 12orbi12i 542 . . . . 5 ((∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (𝑤 = 𝑥𝑤 = 𝑦))
142, 4, 133bitr3ri 290 . . . 4 ((𝑤 = 𝑥𝑤 = 𝑦) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
1514abbii 2726 . . 3 {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16 dfpr2 4143 . . . . 5 {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17 pp0ex 4781 . . . . 5 {∅, {∅}} ∈ V
1816, 17eqeltrri 2685 . . . 4 {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19 equequ2 1940 . . . . . . . 8 (𝑣 = 𝑥 → (𝑤 = 𝑣𝑤 = 𝑥))
20 0inp0 4763 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑧 = {∅})
2119, 20prlem1 997 . . . . . . 7 (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2221alrimdv 1844 . . . . . 6 (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2322spimev 2247 . . . . 5 (𝑧 = ∅ → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
24 orcom 401 . . . . . . . 8 (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)))
25 equequ2 1940 . . . . . . . . 9 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
2620con2i 133 . . . . . . . . 9 (𝑧 = {∅} → ¬ 𝑧 = ∅)
2725, 26prlem1 997 . . . . . . . 8 (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)) → 𝑤 = 𝑣)))
2824, 27syl7bi 244 . . . . . . 7 (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2928alrimdv 1844 . . . . . 6 (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
3029spimev 2247 . . . . 5 (𝑧 = {∅} → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
3123, 30jaoi 393 . . . 4 ((𝑧 = ∅ ∨ 𝑧 = {∅}) → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
3218, 31zfrep4 4707 . . 3 {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
3315, 32eqeltri 2684 . 2 {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)} ∈ V
341, 33eqeltri 2684 1 {𝑥, 𝑦} ∈ V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769 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-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128 This theorem is referenced by:  axpr  4832
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