Theorem zfpair 5382

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 5383. Instead, use zfpair2 5391 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 4601	. 2 ⊢ {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)}
2		19.43 1914	. . . . 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 1880	. . . . 5 ⊢ (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5		0ex 5260	. . . . . . . 8 ⊢ ∅ ∈ V
6	5	isseti 3468	. . . . . . 7 ⊢ ∃𝑧 𝑧 = ∅
7		19.41v 1981	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
8	6, 7	mpbiran 722	. . . . . 6 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9		p0ex 5345	. . . . . . . 8 ⊢ {∅} ∈ V
10	9	isseti 3468	. . . . . . 7 ⊢ ∃𝑧 𝑧 = {∅}
11		19.41v 1981	. . . . . . 7 ⊢ (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ (∃𝑧 𝑧 = {∅} ∧ 𝑤 = 𝑦))
12	10, 11	mpbiran 722	. . . . . 6 ⊢ (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)
13	8, 12	orbi12i 928	. . . . 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 2827	. . 3 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16		dfpr2 4601	. . . . 5 ⊢ {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17		pp0ex 5347	. . . . 5 ⊢ {∅, {∅}} ∈ V
18	16, 17	eqeltrri 2857	. . . 4 ⊢ {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19		equequ2 2058	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑥))
20		0inp0 5319	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑧 = {∅})
21	19, 20	prlem1 1070	. . . . . . 7 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
22	21	alrimdv 1961	. . . . . 6 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
23	22	spimevw 2017	. . . . 5 ⊢ (𝑧 = ∅ → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
24		orcom 884	. . . . . . . 8 ⊢ (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)))
25		equequ2 2058	. . . . . . . . 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 1961	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
30	29	spimevw 2017	. . . . 5 ⊢ (𝑧 = {∅} → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
31	23, 30	jaoi 871	. . . 4 ⊢ ((𝑧 = ∅ ∨ 𝑧 = {∅}) → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
32	18, 31	zfrep4 5245	. . 3 ⊢ {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
33	15, 32	eqeltri 2856	. 2 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} ∈ V
34	1, 33	eqeltri 2856	1 ⊢ {𝑥, 𝑦} ∈ V

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  Vcvv 3450  ∅c0 4278  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-pw 4558  df-sn 4585  df-pr 4587

This theorem is used by:  axprALT  5383