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Theorem zfpair 5032
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 5033. Instead, use zfpair2 5035 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 4334 . 2 {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)}
2 19.43 1962 . . . . 5 (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)))
3 prlem2 1042 . . . . . 6 (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
43exbii 1924 . . . . 5 (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5 0ex 4924 . . . . . . . 8 ∅ ∈ V
65isseti 3361 . . . . . . 7 𝑧 𝑧 = ∅
7 19.41v 2029 . . . . . . 7 (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
86, 7mpbiran 688 . . . . . 6 (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9 p0ex 4984 . . . . . . . 8 {∅} ∈ V
109isseti 3361 . . . . . . 7 𝑧 𝑧 = {∅}
11 19.41v 2029 . . . . . . 7 (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ (∃𝑧 𝑧 = {∅} ∧ 𝑤 = 𝑦))
1210, 11mpbiran 688 . . . . . 6 (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)
138, 12orbi12i 898 . . . . 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 2888 . . 3 {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16 dfpr2 4334 . . . . 5 {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17 pp0ex 4986 . . . . 5 {∅, {∅}} ∈ V
1816, 17eqeltrri 2847 . . . 4 {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19 equequ2 2111 . . . . . . . 8 (𝑣 = 𝑥 → (𝑤 = 𝑣𝑤 = 𝑥))
20 0inp0 4968 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑧 = {∅})
2119, 20prlem1 1041 . . . . . . 7 (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2221alrimdv 2009 . . . . . 6 (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
2322spimev 2421 . . . . 5 (𝑧 = ∅ → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
24 orcom 857 . . . . . . . 8 (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)))
25 equequ2 2111 . . . . . . . . 9 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
2620con2i 136 . . . . . . . . 9 (𝑧 = {∅} → ¬ 𝑧 = ∅)
2725, 26prlem1 1041 . . . . . . . 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 2009 . . . . . 6 (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
3029spimev 2421 . . . . 5 (𝑧 = {∅} → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
3123, 30jaoi 844 . . . 4 ((𝑧 = ∅ ∨ 𝑧 = {∅}) → ∃𝑣𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
3218, 31zfrep4 4913 . . 3 {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
3315, 32eqeltri 2846 . 2 {𝑤 ∣ (𝑤 = 𝑥𝑤 = 𝑦)} ∈ V
341, 33eqeltri 2846 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 834  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  Vcvv 3351  c0 4063  {csn 4316  {cpr 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319
This theorem is referenced by:  axpr  5033
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