Theorem zfpair 5357

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 5358. Instead, use zfpair2 5370 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 4583	. 2 ⊢ {𝑥, 𝑦} = {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)}
2		19.43 1889	. . . . 5 ⊢ (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)))
3		prlem2 1061	. . . . . 6 ⊢ (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
4	3	exbii 1855	. . . . 5 ⊢ (∃𝑧((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
5		0ex 5236	. . . . . . . 8 ⊢ ∅ ∈ V
6	5	isseti 3450	. . . . . . 7 ⊢ ∃𝑧 𝑧 = ∅
7		19.41v 1956	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ (∃𝑧 𝑧 = ∅ ∧ 𝑤 = 𝑥))
8	6, 7	mpbiran 715	. . . . . 6 ⊢ (∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ↔ 𝑤 = 𝑥)
9		p0ex 5320	. . . . . . . 8 ⊢ {∅} ∈ V
10	9	isseti 3450	. . . . . . 7 ⊢ ∃𝑧 𝑧 = {∅}
11		19.41v 1956	. . . . . . 7 ⊢ (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ (∃𝑧 𝑧 = {∅} ∧ 𝑤 = 𝑦))
12	10, 11	mpbiran 715	. . . . . 6 ⊢ (∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦) ↔ 𝑤 = 𝑦)
13	8, 12	orbi12i 920	. . . . 5 ⊢ ((∃𝑧(𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ ∃𝑧(𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
14	2, 4, 13	3bitr3ri 303	. . . 4 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) ↔ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦))))
15	14	abbii 2807	. . 3 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} = {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))}
16		dfpr2 4583	. . . . 5 ⊢ {∅, {∅}} = {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})}
17		pp0ex 5322	. . . . 5 ⊢ {∅, {∅}} ∈ V
18	16, 17	eqeltrri 2837	. . . 4 ⊢ {𝑧 ∣ (𝑧 = ∅ ∨ 𝑧 = {∅})} ∈ V
19		equequ2 2033	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑥))
20		0inp0 5294	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑧 = {∅})
21	19, 20	prlem1 1060	. . . . . . 7 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
22	21	alrimdv 1936	. . . . . 6 ⊢ (𝑣 = 𝑥 → (𝑧 = ∅ → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
23	22	spimevw 1992	. . . . 5 ⊢ (𝑧 = ∅ → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
24		orcom 876	. . . . . . . 8 ⊢ (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) ↔ ((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)))
25		equequ2 2033	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑦))
26	20	con2i 139	. . . . . . . . 9 ⊢ (𝑧 = {∅} → ¬ 𝑧 = ∅)
27	25, 26	prlem1 1060	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = {∅} ∧ 𝑤 = 𝑦) ∨ (𝑧 = ∅ ∧ 𝑤 = 𝑥)) → 𝑤 = 𝑣)))
28	24, 27	syl7bi 256	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → (((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
29	28	alrimdv 1936	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑧 = {∅} → ∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣)))
30	29	spimevw 1992	. . . . 5 ⊢ (𝑧 = {∅} → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
31	23, 30	jaoi 863	. . . 4 ⊢ ((𝑧 = ∅ ∨ 𝑧 = {∅}) → ∃𝑣∀𝑤(((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑣))
32	18, 31	zfrep4 5222	. . 3 ⊢ {𝑤 ∣ ∃𝑧((𝑧 = ∅ ∨ 𝑧 = {∅}) ∧ ((𝑧 = ∅ ∧ 𝑤 = 𝑥) ∨ (𝑧 = {∅} ∧ 𝑤 = 𝑦)))} ∈ V
33	15, 32	eqeltri 2836	. 2 ⊢ {𝑤 ∣ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)} ∈ V
34	1, 33	eqeltri 2836	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718  Vcvv 3432  ∅c0 4268  {csn 4562  {cpr 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-pw 4538  df-sn 4563  df-pr 4565

This theorem is referenced by:  axprALT  5358