Theorem wlogle 11670

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 1172	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → (𝜒 ↔ 𝜃))
7	4, 6	mpbid 232	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜃)
8	1, 2, 3, 7, 4	wloglei 11669	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113   ⊆ wss 3901   class class class wbr 5098  ℝcr 11025   ≤ cle 11167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-pre-lttri 11100

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172

This theorem is referenced by:  vdwlem12  16920  iundisj2  25506  volcn  25563  dvlip  25954  ftc1a  26000  iundisj2f  32665  iundisj2fi  32877  erdszelem9  35393  ftc1anc  37898