Theorem preleqALT 9608

Description: Alternate proof of preleq 9607, not based on preleqg 9606: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
preleq.b	⊢ 𝐵 ∈ V
preleqALT.d	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
preleqALT	⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem preleqALT

Step	Hyp	Ref	Expression
1		preleq.b	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
2	1	jctr 524	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
3		preleqALT.d	. . . . . . . . . 10 ⊢ 𝐷 ∈ V
4	3	jctr 524	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐷 → (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ V))
5		preq12bg 4846	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
6	2, 4, 5	syl2an 595	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
7	6	biimpa 476	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
8	7	ord 861	. . . . . 6 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
9		en2lp 9597	. . . . . . 7 ⊢ ¬ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷)
10		eleq12 2815	. . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ∈ 𝐵 ↔ 𝐷 ∈ 𝐶))
11	10	anbi1d 629	. . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ↔ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷)))
12	9, 11	mtbiri 327	. . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷))
13	8, 12	syl6 35	. . . . 5 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷)))
14	13	con4d 115	. . . 4 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
15	14	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
16	15	pm2.43a 54	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
17	16	imp 406	1 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  Vcvv 3466  {cpr 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-reg 9583

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-eprel 5570  df-fr 5621

This theorem is referenced by: (None)