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Theorem wlogle 10546
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 1220 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → (𝜒𝜃))
74, 6mpbid 222 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)
81, 2, 3, 7, 4wloglei 10545 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1988  wss 3567   class class class wbr 4644  cr 9920  cle 10060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-resscn 9978  ax-pre-lttri 9995
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065
This theorem is referenced by:  vdwlem12  15677  iundisj2  23298  volcn  23355  dvlip  23737  ftc1a  23781  iundisj2f  29375  iundisj2fi  29530  erdszelem9  31155  ftc1anc  33464
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