Theorem wlogle 11778

Description: If the predicate 𝜒(𝑥, 𝑦) is symmetric under interchange of 𝑥, 𝑦, then "without loss of generality" we can assume that 𝑥 ≤ 𝑦. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)

Hypotheses

Ref	Expression
wlogle.1	⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒))
wlogle.2	⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
wlogle.3	⊢ (𝜑 → 𝑆 ⊆ ℝ)
wlogle.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝜒 ↔ 𝜃))
wlogle.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒)

Assertion

Ref	Expression
wlogle	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝜑   𝑤,𝑆,𝑥,𝑦,𝑧   𝜓,𝑥,𝑦   𝜒,𝑤,𝑧

Allowed substitution hints:   𝜓(𝑧,𝑤)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wlogle

Step	Hyp	Ref	Expression
1		wlogle.1	. 2 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒))
2		wlogle.2	. 2 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))
3		wlogle.3	. 2 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
4		wlogle.5	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒)
5		wlogle.4	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝜒 ↔ 𝜃))
6	5	3adantr3 1169	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → (𝜒 ↔ 𝜃))
7	4, 6	mpbid 231	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜃)
8	1, 2, 3, 7, 4	wloglei 11777	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2099   ⊆ wss 3947   class class class wbr 5148  ℝcr 11138   ≤ cle 11280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-resscn 11196  ax-pre-lttri 11213

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285

This theorem is referenced by:  vdwlem12  16961  iundisj2  25491  volcn  25548  dvlip  25939  ftc1a  25985  iundisj2f  32393  iundisj2fi  32578  erdszelem9  34809  ftc1anc  37174