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Theorem zfpair 4902
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 4903. Instead, use zfpair2 4905 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 4193 . 2 {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)}
2 19.43 1809 . . . . 5 (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)))
3 prlem2 1006 . . . . . 6 (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
43exbii 1773 . . . . 5 (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5 0ex 4788 . . . . . . . 8 ∅ ∈ V
65isseti 3207 . . . . . . 7 𝑧 𝑧 = ∅
7 19.41v 1913 . . . . . . 7 (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
86, 7mpbiran 953 . . . . . 6 (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9 p0ex 4851 . . . . . . . 8 {∅} ∈ V
109isseti 3207 . . . . . . 7 𝑧 𝑧 = {∅}
11 19.41v 1913 . . . . . . 7 (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ (∃𝑧 𝑧 = {∅} ∧ 𝑤 = 𝑦))
1210, 11mpbiran 953 . . . . . 6 (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)
138, 12orbi12i 543 . . . . 5 ((∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (𝑤 = 𝑥𝑤 = 𝑦))
142, 4, 133bitr3ri 291 . . . 4 ((𝑤 = 𝑥𝑤 = 𝑦) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
1514abbii 2738 . . 3 {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16 dfpr2 4193 . . . . 5 {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17 pp0ex 4853 . . . . 5 {∅, {∅}} ∈ V
1816, 17eqeltrri 2697 . . . 4 {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19 equequ2 1952 . . . . . . . 8 (𝑣 = 𝑥 → (𝑤 = 𝑣𝑤 = 𝑥))
20 0inp0 4835 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑧 = {∅})
2119, 20prlem1 1005 . . . . . . 7 (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2221alrimdv 1856 . . . . . 6 (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2322spimev 2258 . . . . 5 (𝑧 = ∅ → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
24 orcom 402 . . . . . . . 8 (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)))
25 equequ2 1952 . . . . . . . . 9 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
2620con2i 134 . . . . . . . . 9 (𝑧 = {∅} → ¬ 𝑧 = ∅)
2725, 26prlem1 1005 . . . . . . . 8 (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)) → 𝑤 = 𝑣)))
2824, 27syl7bi 245 . . . . . . 7 (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2928alrimdv 1856 . . . . . 6 (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
3029spimev 2258 . . . . 5 (𝑧 = {∅} → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
3123, 30jaoi 394 . . . 4 ((𝑧 = ∅ ∨ 𝑧 = {∅}) → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
3218, 31zfrep4 4777 . . 3 {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
3315, 32eqeltri 2696 . 2 {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)} ∈ V
341, 33eqeltri 2696 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  wal 1480   = wceq 1482  wex 1703  wcel 1989  {cab 2607  Vcvv 3198  c0 3913  {csn 4175  {cpr 4177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-pw 4158  df-sn 4176  df-pr 4178
This theorem is referenced by:  axpr  4903
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