Theorem zfpair 5394

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 axprALT 5395. Instead, use zfpair2 5407 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.

Step	Hyp	Ref	Expression
1		dfpr2 4611	. 2 ⊢ {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)}
2		19.43 1912	. . . . 5 ⊢ (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)))
3		prlem2 1071	. . . . . 6 ⊢ (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
4	3	exbii 1878	. . . . 5 ⊢ (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5		0ex 5271	. . . . . . . 8 ⊢ ∅ ∈ V
6	5	isseti 3473	. . . . . . 7 ⊢ ∃𝑧 𝑧 = ∅
7		19.41v 1979	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
8	6, 7	mpbiran 721	. . . . . 6 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9		p0ex 5357	. . . . . . . 8 ⊢ {∅} ∈ V
10	9	isseti 3473	. . . . . . 7 ⊢ ∃𝑧 𝑧 = {∅}
11		19.41v 1979	. . . . . . 7 ⊢ (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ (∃𝑧 𝑧 = {∅} ∧ 𝑤 = 𝑦))
12	10, 11	mpbiran 721	. . . . . 6 ⊢ (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)
13	8, 12	orbi12i 927	. . . . 5 ⊢ ((∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
14	2, 4, 13	3bitr3ri 305	. . . 4 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
15	14	abbii 2830	. . 3 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16		dfpr2 4611	. . . . 5 ⊢ {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17		pp0ex 5359	. . . . 5 ⊢ {∅, {∅}} ∈ V
18	16, 17	eqeltrri 2860	. . . 4 ⊢ {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19		equequ2 2056	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑥))
20		0inp0 5331	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑧 = {∅})
21	19, 20	prlem1 1070	. . . . . . 7 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
22	21	alrimdv 1959	. . . . . 6 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
23	22	spimevw 2015	. . . . 5 ⊢ (𝑧 = ∅ → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
24		orcom 883	. . . . . . . 8 ⊢ (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)))
25		equequ2 2056	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑦))
26	20	con2i 140	. . . . . . . . 9 ⊢ (𝑧 = {∅} → ¬ 𝑧 = ∅)
27	25, 26	prlem1 1070	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)) → 𝑤 = 𝑣)))
28	24, 27	syl7bi 258	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
29	28	alrimdv 1959	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
30	29	spimevw 2015	. . . . 5 ⊢ (𝑧 = {∅} → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
31	23, 30	jaoi 870	. . . 4 ⊢ ((𝑧 = ∅ ∨ 𝑧 = {∅}) → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
32	18, 31	zfrep4 5255	. . 3 ⊢ {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
33	15, 32	eqeltri 2859	. 2 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} ∈ V
34	1, 33	eqeltri 2859	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455  ∅c0 4287  {csn 4590  {cpr 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-pw 4565  df-sn 4591  df-pr 4593

This theorem is referenced by:  axprALT  5395