Theorem zfpair 5383

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 5384. Instead, use zfpair2 5392 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 4605	. 2 ⊢ {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)}
2		19.43 1915	. . . . 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 1881	. . . . 5 ⊢ (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5		0ex 5261	. . . . . . . 8 ⊢ ∅ ∈ V
6	5	isseti 3469	. . . . . . 7 ⊢ ∃𝑧 𝑧 = ∅
7		19.41v 1982	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
8	6, 7	mpbiran 722	. . . . . 6 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9		p0ex 5346	. . . . . . . 8 ⊢ {∅} ∈ V
10	9	isseti 3469	. . . . . . 7 ⊢ ∃𝑧 𝑧 = {∅}
11		19.41v 1982	. . . . . . 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 2828	. . 3 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16		dfpr2 4605	. . . . 5 ⊢ {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17		pp0ex 5348	. . . . 5 ⊢ {∅, {∅}} ∈ V
18	16, 17	eqeltrri 2858	. . . 4 ⊢ {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19		equequ2 2059	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑥))
20		0inp0 5320	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑧 = {∅})
21	19, 20	prlem1 1070	. . . . . . 7 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
22	21	alrimdv 1962	. . . . . 6 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
23	22	spimevw 2018	. . . . 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 2059	. . . . . . . . 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 1962	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
30	29	spimevw 2018	. . . . 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 5246	. . 3 ⊢ {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
33	15, 32	eqeltri 2857	. 2 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} ∈ V
34	1, 33	eqeltri 2857	1 ⊢ {𝑥, 𝑦} ∈ V

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  Vcvv 3451  ∅c0 4279  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-pw 4559  df-sn 4585  df-pr 4587

This theorem is used by:  axprALT  5384