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Theorem wlogle 10851
Description: If the predicate 𝜒(𝑥, 𝑦) is symmetric under interchange of 𝑥, 𝑦, then "without loss of generality" we can assume that 𝑥𝑦. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)
Hypotheses
Ref Expression
wlogle.1 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
wlogle.2 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
wlogle.3 (𝜑𝑆 ⊆ ℝ)
wlogle.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝜒𝜃))
wlogle.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
Assertion
Ref Expression
wlogle ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 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
StepHypRef Expression
1 wlogle.1 . 2 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))
2 wlogle.2 . 2 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))
3 wlogle.3 . 2 (𝜑𝑆 ⊆ ℝ)
4 wlogle.5 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)
5 wlogle.4 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝜒𝜃))
653adantr3 1213 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → (𝜒𝜃))
74, 6mpbid 224 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
81, 2, 3, 7, 4wloglei 10850 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  wss 3767   class class class wbr 4841  cr 10221  cle 10362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-resscn 10279  ax-pre-lttri 10296
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367
This theorem is referenced by:  vdwlem12  16026  iundisj2  23654  volcn  23711  dvlip  24094  ftc1a  24138  iundisj2f  29912  iundisj2fi  30066  erdszelem9  31690  ftc1anc  33973
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